Network Theorems, Linearity, and Equivalence

U04-L01 · Superposition of Independent Sources

Unit: 4 — Network Theorems Core relation: x=x_1+x_2 Essential ideas: linearity, source deactivation, and dependent source retention.

Learning objectives

After this lesson, explain superposition of independent sources and its circuit context, assign reference polarities and directions, derive or use x=x_1+x_2 with SI units, solve a numerical example, and evaluate its validity using conservation laws and a practical limiting case.

Physical and mathematical foundation

Superposition applies to current and voltage, but not directly to dissipated power. 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 x=x_1+x_2 requires knowing both the connected terminals and the applicable ideal-component assumptions.

Within network theorems, 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 — Linearity. Identify the branch or terminals to which this idea applies, state its units, and connect it with x=x_1+x_2. 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 — Source deactivation. Identify the branch or terminals to which this idea applies, state its units, and connect it with x=x_1+x_2. 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 dependent source retention. Identify the branch or terminals to which this idea applies, state its units, and connect it with x=x_1+x_2. 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 linear resistor carries 0.3 A due to source A and −0.1 A due to source B.

Result. The combined signed current is 0.2 A, not 0.4 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. Superposition applies to current and voltage, but not directly to dissipated power. 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

1. Draw the topology, identify nodes, and choose polarities and branch-current arrows; write down which source and element quantities are known.
2. Combine the component relation x=x_1+x_2 with KCL, KVL, and state continuity where relevant; solve symbolically before substituting numbers.
3. Convert units, calculate, and cross-check using a conservation equation or limiting case. Explain why the result is physically plausible.
4. Identify the dominant idealization. Replace an ideal independent voltage source by a short and an ideal current source by an open when deactivating. 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 linearity 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 `x=x_1+x_2` remains valid and which previous shortcut fails.

Key limitation: Replace an ideal independent voltage source by a short and an ideal current source by an open when deactivating.

U04-L02 · Thévenin Equivalent Circuits

Unit: 4 — Network Theorems Core relation: V_TH=V_OC Essential ideas: open-circuit voltage and driving-point resistance.

Learning objectives

After this lesson, explain thévenin equivalent circuits and its circuit context, assign reference polarities and directions, derive or use V_TH=V_OC with SI units, solve a numerical example, and evaluate its validity using conservation laws and a practical limiting case.

Physical and mathematical foundation

An equivalent circuit reproduces terminal behavior for an external load. 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_TH=V_OC requires knowing both the connected terminals and the applicable ideal-component assumptions.

Within network theorems, 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 — Open-circuit voltage and driving-point resistance. Identify the branch or terminals to which this idea applies, state its units, and connect it with V_TH=V_OC. 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 network measures 9 V open-circuit and has a 3 Ω equivalent resistance.

Result. Its Thévenin equivalent is a 9 V ideal source in series with 3 Ω. 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 equivalent circuit reproduces terminal behavior for an external load. 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

1. Draw the topology, identify nodes, and choose polarities and branch-current arrows; write down which source and element quantities are known.
2. Combine the component relation V_TH=V_OC with KCL, KVL, and state continuity where relevant; solve symbolically before substituting numbers.
3. Convert units, calculate, and cross-check using a conservation equation or limiting case. Explain why the result is physically plausible.
4. Identify the dominant idealization. If dependent sources are present, use a test source or a suitable open/short measurement to find resistance. 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 open-circuit voltage and driving-point resistance 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_TH=V_OC` remains valid and which previous shortcut fails.

Key limitation: If dependent sources are present, use a test source or a suitable open/short measurement to find resistance.

U04-L03 · Norton Equivalent Circuits

Unit: 4 — Network Theorems Core relation: I_N=I_SC=V_TH/R_TH Essential ideas: short-circuit current, parallel resistance, and conversion.

Learning objectives

After this lesson, explain norton equivalent circuits and its circuit context, assign reference polarities and directions, derive or use I_N=I_SC=V_TH/R_TH with SI units, solve a numerical example, and evaluate its validity using conservation laws and a practical limiting case.

Physical and mathematical foundation

Thévenin and Norton are dual two-terminal descriptions of the same linear network. 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 I_N=I_SC=V_TH/R_TH requires knowing both the connected terminals and the applicable ideal-component assumptions.

Within network theorems, 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 — Short-circuit current. Identify the branch or terminals to which this idea applies, state its units, and connect it with I_N=I_SC=V_TH/R_TH. 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 resistance. Identify the branch or terminals to which this idea applies, state its units, and connect it with I_N=I_SC=V_TH/R_TH. 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 conversion. Identify the branch or terminals to which this idea applies, state its units, and connect it with I_N=I_SC=V_TH/R_TH. 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 9 V Thévenin source with 3 Ω series resistance is converted to Norton form.

Result. The Norton source is 3 A in parallel with 3 Ω. 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. Thévenin and Norton are dual two-terminal descriptions of the same linear network. 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

1. Draw the topology, identify nodes, and choose polarities and branch-current arrows; write down which source and element quantities are known.
2. Combine the component relation I_N=I_SC=V_TH/R_TH with KCL, KVL, and state continuity where relevant; solve symbolically before substituting numbers.
3. Convert units, calculate, and cross-check using a conservation equation or limiting case. Explain why the result is physically plausible.
4. Identify the dominant idealization. Test whether a short-circuit condition is mathematically and physically admissible. 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 short-circuit current 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 `I_N=I_SC=V_TH/R_TH` remains valid and which previous shortcut fails.

Key limitation: Test whether a short-circuit condition is mathematically and physically admissible.

U04-L04 · Maximum Power Transfer

Unit: 4 — Network Theorems Core relation: P_L=V_TH²R_L/(R_TH+R_L)² Essential ideas: load matching, power-vs-resistance curve, and efficiency.

Learning objectives

After this lesson, explain maximum power transfer and its circuit context, assign reference polarities and directions, derive or use P_L=V_TH²R_L/(R_TH+R_L)² with SI units, solve a numerical example, and evaluate its validity using conservation laws and a practical limiting case.

Physical and mathematical foundation

Matching maximizes load power for a positive fixed source resistance, not overall efficiency. 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 P_L=V_TH²R_L/(R_TH+R_L)² requires knowing both the connected terminals and the applicable ideal-component assumptions.

Within network theorems, 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 — Load matching. Identify the branch or terminals to which this idea applies, state its units, and connect it with P_L=V_TH²R_L/(R_TH+R_L)². 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 — Power-vs-resistance curve. Identify the branch or terminals to which this idea applies, state its units, and connect it with P_L=V_TH²R_L/(R_TH+R_L)². 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 efficiency. Identify the branch or terminals to which this idea applies, state its units, and connect it with P_L=V_TH²R_L/(R_TH+R_L)². 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 12 V Thévenin source with 6 Ω internal resistance drives a matched load.

Result. At R_L=6 Ω the load receives 6 W, while the internal resistance also dissipates 6 W. 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. Matching maximizes load power for a positive fixed source resistance, not overall efficiency. 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

1. Draw the topology, identify nodes, and choose polarities and branch-current arrows; write down which source and element quantities are known.
2. Combine the component relation P_L=V_TH²R_L/(R_TH+R_L)² with KCL, KVL, and state continuity where relevant; solve symbolically before substituting numbers.
3. Convert units, calculate, and cross-check using a conservation equation or limiting case. Explain why the result is physically plausible.
4. Identify the dominant idealization. Do not choose maximum-power matching by default in power distribution or battery systems. 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 load matching 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 `P_L=V_TH²R_L/(R_TH+R_L)²` remains valid and which previous shortcut fails.

Key limitation: Do not choose maximum-power matching by default in power distribution or battery systems.

U04-L05 · Network Equivalence and Sensitivity

Unit: 4 — Network Theorems Core relation: V_L=V_TH R_L/(R_TH+R_L) Essential ideas: substitution, load sweeps, source resistance, and model validation.

Learning objectives

After this lesson, explain network equivalence and sensitivity and its circuit context, assign reference polarities and directions, derive or use V_L=V_TH R_L/(R_TH+R_L) with SI units, solve a numerical example, and evaluate its validity using conservation laws and a practical limiting case.

Physical and mathematical foundation

A single load point does not by itself prove two models are equivalent for arbitrary loads. 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=V_TH R_L/(R_TH+R_L) requires knowing both the connected terminals and the applicable ideal-component assumptions.

Within network theorems, 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 — Substitution. Identify the branch or terminals to which this idea applies, state its units, and connect it with V_L=V_TH R_L/(R_TH+R_L). 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 — Load sweeps. Identify the branch or terminals to which this idea applies, state its units, and connect it with V_L=V_TH R_L/(R_TH+R_L). 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 — Source resistance. Identify the branch or terminals to which this idea applies, state its units, and connect it with V_L=V_TH R_L/(R_TH+R_L). 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 10 V source with 2 Ω source resistance drives an 8 Ω load.

Result. The load voltage is 8 V, and the load dissipates 8 W. 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. A single load point does not by itself prove two models are equivalent for arbitrary loads. 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

1. Draw the topology, identify nodes, and choose polarities and branch-current arrows; write down which source and element quantities are known.
2. Combine the component relation V_L=V_TH R_L/(R_TH+R_L) with KCL, KVL, and state continuity where relevant; solve symbolically before substituting numbers.
3. Convert units, calculate, and cross-check using a conservation equation or limiting case. Explain why the result is physically plausible.
4. Identify the dominant idealization. Compare terminal behavior across several loads and explicitly retain the sign conventions. 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 substitution 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=V_TH R_L/(R_TH+R_L)` remains valid and which previous shortcut fails.

Key limitation: Compare terminal behavior across several loads and explicitly retain the sign conventions.