PE Exam Exam > PE Exam Notes > Electrical & Computer Engineering for PE > Cheatsheet: Network Theorems
Cheatsheet: Network Theorems
1. Superposition Theorem
1.1 Principle
| Concept | Description |
|---|---|
| Statement | In a linear circuit with multiple independent sources, the voltage or current in any element equals the algebraic sum of voltages or currents produced by each source acting independently |
| Linearity Requirement | Applies only to linear circuits; all resistors, capacitors, inductors must have constant values |
| Source Deactivation | Voltage sources → short circuit (0 V); Current sources → open circuit (0 A) |
1.2 Procedure
- Deactivate all independent sources except one
- Calculate the desired voltage or current due to that single source
- Repeat for each independent source individually
- Algebraically sum all individual responses (consider polarity/direction)
- Dependent sources remain active throughout all calculations
1.3 Power Calculation
- Cannot apply superposition directly to power (nonlinear relationship: P = I²R or P = V²/R)
- Calculate total voltage or current first, then compute power: P = VI or P = I²R
2. Thévenin's Theorem
2.1 Principle
| Parameter | Definition |
|---|---|
| Statement | Any linear two-terminal network can be replaced by an equivalent circuit consisting of a voltage source VTH in series with a resistance RTH |
| VTH | Open-circuit voltage across the terminals when load is removed |
| RTH | Equivalent resistance looking into the terminals with all independent sources deactivated |
2.2 Determining VTH
- Remove the load from the terminals
- Calculate the open-circuit voltage VOC across the terminals
- VTH = VOC
2.3 Determining RTH
2.3.1 Method 1: Direct Calculation
- Remove the load
- Deactivate all independent sources (short voltage sources, open current sources)
- Calculate resistance looking back into the circuit from the terminals
- Dependent sources remain active but do not contribute without excitation
2.3.2 Method 2: Short-Circuit Current
- Calculate short-circuit current ISC when terminals are shorted
- RTH = VTH / ISC = VOC / ISC
- Use when circuit contains dependent sources
2.3.3 Method 3: Test Source
- Apply a test voltage VT or test current IT at the terminals
- Calculate resulting current IT or voltage VT
- RTH = VT / IT
- Deactivate all independent sources; keep dependent sources active
2.4 Applications
- Simplifies analysis when varying a single load element
- Maximum power transfer analysis
- Circuit design and troubleshooting
3. Norton's Theorem
3.1 Principle
| Parameter | Definition |
|---|---|
| Statement | Any linear two-terminal network can be replaced by an equivalent circuit consisting of a current source IN in parallel with a resistance RN |
| IN | Short-circuit current through the terminals when load is replaced by a short circuit |
| RN | Equivalent resistance looking into the terminals with all independent sources deactivated; RN = RTH |
3.2 Determining IN
- Short-circuit the terminals where load was connected
- Calculate the short-circuit current ISC flowing through the short
- IN = ISC
3.3 Determining RN
- Same methods as RTH (RN = RTH)
- Alternatively: RN = VOC / ISC
3.4 Thévenin-Norton Conversion
| Conversion | Relationships |
|---|---|
| Thévenin to Norton | IN = VTH / RTH; RN = RTH |
| Norton to Thévenin | VTH = IN × RN; RTH = RN |
4. Maximum Power Transfer Theorem
4.1 DC Circuits
| Concept | Description |
|---|---|
| Condition | Maximum power is transferred to the load when load resistance RL = RTH |
| Maximum Power | Pmax = VTH² / (4RTH) |
| Efficiency | η = 50% at maximum power transfer (half dissipated in source resistance) |
| Load Voltage | VL = VTH / 2 when RL = RTH |
| Load Current | IL = VTH / (2RTH) when RL = RTH |
4.2 AC Circuits
| Load Type | Condition for Maximum Power |
|---|---|
| Resistive Load | RL = |ZTH| (magnitude of Thévenin impedance) |
| Complex Load | ZL = ZTH* (complex conjugate of Thévenin impedance) |
| Maximum Power (Complex) | Pmax = |VTH|² / (4RTH) where RTH = Re{ZTH} |
4.3 Key Points
- If ZTH = RTH + jXTH, then ZL = RTH - jXTH for maximum power
- Reactive components in load cancel reactive components in source
- Practical systems operate at higher efficiency, not maximum power transfer
5. Source Transformation
5.1 Principle
| Transformation | Description |
|---|---|
| Voltage to Current | Voltage source VS in series with R → Current source IS = VS/R in parallel with R |
| Current to Voltage | Current source IS in parallel with R → Voltage source VS = IS × R in series with R |
| Resistance | Resistance value R remains the same in both representations |
5.2 Applications
- Simplify circuits by converting between source types
- Combine parallel current sources or series voltage sources
- Cannot transform ideal sources (R = 0 for ideal voltage source, R = ∞ for ideal current source)
- Polarity: positive terminal of VS corresponds to direction of IS leaving the source
6. Millman's Theorem
6.1 Principle
| Concept | Description |
|---|---|
| Statement | For parallel branches with voltage sources and series resistances, the common voltage across all branches can be calculated directly |
| Formula | V = (V₁/R₁ + V₂/R₂ + ... + Vn/Rn) / (1/R₁ + 1/R₂ + ... + 1/Rn) |
| Alternative Form | V = (V₁G₁ + V₂G₂ + ... + VnGn) / (G₁ + G₂ + ... + Gn) where G = 1/R |
6.2 Procedure
- Identify all parallel branches with voltage sources and series resistances
- Convert each branch to equivalent conductance G = 1/R
- Apply formula to find common node voltage
- Calculate branch currents using Ohm's law
7. Substitution Theorem
7.1 Principle
| Concept | Description |
|---|---|
| Statement | Any branch in a network with voltage V and current I can be replaced by a voltage source of value V or a current source of value I without affecting the rest of the circuit |
| Voltage Source Replacement | Replace branch with voltage source equal to the voltage across the branch (maintain polarity) |
| Current Source Replacement | Replace branch with current source equal to the current through the branch (maintain direction) |
7.2 Applications
- Simplify analysis by replacing complex branches
- Isolate portions of circuits for separate analysis
- Must know voltage and current in branch before substitution
8. Reciprocity Theorem
8.1 Principle
| Concept | Description |
|---|---|
| Statement | In a linear bilateral network, if a voltage source V in branch A produces current I in branch B, then the same voltage source V in branch B will produce the same current I in branch A |
| Bilateral Network | Circuit elements have the same characteristics regardless of current direction (resistors, inductors, capacitors) |
| Limitation | Does not apply to networks with dependent sources or nonlinear elements |
8.2 Applications
- Verify circuit calculations by interchanging source and response locations
- Simplify measurements in testing (swap excitation and measurement points)
- Antenna theory: transmitting and receiving patterns are identical
9. Compensation Theorem
9.1 Principle
| Concept | Description |
|---|---|
| Statement | If impedance Z in a branch changes by ΔZ, the change in current throughout the network can be calculated by inserting a voltage source V = -I × ΔZ in series with the changed branch |
| Change in Current | ΔI = change in current due to impedance change ΔZ |
| Compensation Voltage | Vcomp = -Ioriginal × ΔZ (opposes original current direction) |
9.2 Applications
- Analyze effects of component tolerance variations
- Determine sensitivity of circuits to parameter changes
- Calculate changes without re-analyzing entire circuit
10. Tellegen's Theorem
10.1 Principle
| Concept | Description |
|---|---|
| Statement | For any lumped network satisfying Kirchhoff's laws, the sum of instantaneous powers in all branches equals zero: Σ(vk × ik) = 0 |
| Interpretation | Total power delivered equals total power absorbed in any network |
| Applicability | Applies to linear and nonlinear networks, time-varying or time-invariant |
10.2 Key Points
- Based solely on topology (KCL and KVL), not element characteristics
- Valid for AC and DC circuits
- Used in power conservation verification and network analysis validation
11. Application Guidelines
11.1 Theorem Selection
| Use Case | Recommended Theorem |
|---|---|
| Multiple sources, finding one response | Superposition |
| Varying single load, multiple calculations | Thévenin or Norton |
| Maximum power to load | Maximum Power Transfer |
| Parallel voltage sources | Millman's Theorem |
| Mixing series and parallel sources | Source Transformation |
11.2 Common Mistakes
- Applying superposition to power calculations (must use voltage/current first)
- Forgetting to keep dependent sources active during Thévenin/Norton calculation
- Incorrect source deactivation (voltage source → short, current source → open)
- Confusing maximum power transfer with maximum efficiency
- Attempting to transform ideal sources without series/parallel resistance
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Apr 20, 2026 Last updated
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