PE Exam Exam > PE Exam Notes > Electrical & Computer Engineering for PE > Cheatsheet: DC Circuits
Cheatsheet: DC Circuits
1. Fundamental Laws
1.1 Ohm's Law
| Formula | Description |
|---|---|
| V = IR | Voltage (V) equals current (I) times resistance (R) |
| I = V/R | Current equals voltage divided by resistance |
| R = V/I | Resistance equals voltage divided by current |
1.2 Kirchhoff's Current Law (KCL)
| Law | Statement |
|---|---|
| ΣI = 0 | Sum of currents entering a node equals sum of currents leaving the node |
| Application | I₁ + I₂ + I₃ + ... + Iₙ = 0 (sign convention: entering positive, leaving negative) |
1.3 Kirchhoff's Voltage Law (KVL)
| Law | Statement |
|---|---|
| ΣV = 0 | Sum of voltage drops around any closed loop equals zero |
| Application | V₁ + V₂ + V₃ + ... + Vₙ = 0 (sign convention: voltage rise positive, voltage drop negative) |
2. Power and Energy
2.1 Power Formulas
| Formula | Variables |
|---|---|
| P = VI | Power (W) = Voltage (V) × Current (A) |
| P = I²R | Power dissipated in resistor |
| P = V²/R | Power dissipated in resistor |
2.2 Energy
| Formula | Description |
|---|---|
| W = Pt | Energy (J) = Power (W) × time (s) |
| W = VIt | Energy from voltage and current |
3. Resistor Combinations
3.1 Series Resistors
| Property | Formula/Description |
|---|---|
| Total Resistance | R_total = R₁ + R₂ + R₃ + ... + Rₙ |
| Current | Same current flows through all resistors: I_total = I₁ = I₂ = I₃ |
| Voltage | V_total = V₁ + V₂ + V₃ + ... + Vₙ |
| Voltage Division | Vₙ = V_total × (Rₙ/R_total) |
3.2 Parallel Resistors
| Property | Formula/Description |
|---|---|
| Total Resistance | 1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + ... + 1/Rₙ |
| Two Resistors | R_total = (R₁R₂)/(R₁ + R₂) = Product/Sum |
| N Equal Resistors | R_total = R/N |
| Voltage | Same voltage across all resistors: V_total = V₁ = V₂ = V₃ |
| Current | I_total = I₁ + I₂ + I₃ + ... + Iₙ |
| Current Division | Iₙ = I_total × (R_total/Rₙ) |
3.3 Series-Parallel Combinations
- Identify series and parallel sections
- Reduce parallel sections first using parallel formula
- Combine series resistances
- Repeat until circuit is fully simplified
4. Network Analysis Methods
4.1 Nodal Analysis
| Step | Procedure |
|---|---|
| 1 | Select reference node (ground, V = 0) |
| 2 | Label remaining node voltages (V₁, V₂, ...) |
| 3 | Apply KCL at each non-reference node |
| 4 | Express currents in terms of node voltages using Ohm's Law |
| 5 | Solve system of equations for node voltages |
4.2 Mesh Analysis
| Step | Procedure |
|---|---|
| 1 | Identify all independent meshes (loops) |
| 2 | Assign mesh currents (clockwise convention) |
| 3 | Apply KVL around each mesh |
| 4 | Express voltages in terms of mesh currents using Ohm's Law |
| 5 | Solve system of equations for mesh currents |
4.3 Superposition Theorem
| Step | Procedure |
|---|---|
| 1 | Turn off all sources except one (voltage sources → short circuit, current sources → open circuit) |
| 2 | Calculate response due to that single source |
| 3 | Repeat for each independent source |
| 4 | Sum all individual responses to get total response |
| Limitation | Only applies to linear circuits; valid for voltage and current, NOT power |
5. Network Theorems
5.1 Thévenin's Theorem
| Parameter | Description |
|---|---|
| V_TH | Open-circuit voltage across load terminals |
| R_TH | Equivalent resistance seen from load terminals with all sources deactivated |
| Circuit | Single voltage source V_TH in series with resistor R_TH |
- Finding R_TH: Turn off independent sources, calculate resistance at terminals
- Alternative R_TH: R_TH = V_OC/I_SC (open-circuit voltage / short-circuit current)
5.2 Norton's Theorem
| Parameter | Description |
|---|---|
| I_N | Short-circuit current through load terminals |
| R_N | Equivalent resistance seen from load terminals with all sources deactivated (R_N = R_TH) |
| Circuit | Single current source I_N in parallel with resistor R_N |
5.3 Source Transformation
| From | To |
|---|---|
| Voltage source V_S in series with R | Current source I_S = V_S/R in parallel with R |
| Current source I_S in parallel with R | Voltage source V_S = I_S×R in series with R |
- Thévenin to Norton: I_N = V_TH/R_TH, R_N = R_TH
- Norton to Thévenin: V_TH = I_N×R_N, R_TH = R_N
5.4 Maximum Power Transfer Theorem
| Condition | Result |
|---|---|
| R_Load = R_TH | Maximum power transferred to load |
| P_max | P_max = V²_TH/(4R_TH) |
| Efficiency | 50% at maximum power transfer |
6. Voltage and Current Sources
6.1 Independent Sources
| Type | Characteristics |
|---|---|
| Ideal Voltage Source | Maintains constant voltage regardless of current; zero internal resistance |
| Ideal Current Source | Maintains constant current regardless of voltage; infinite internal resistance |
| Practical Voltage Source | V_terminal = V_S - IR_S (R_S is internal resistance) |
| Practical Current Source | I_terminal = I_S - V/R_P (R_P is internal resistance) |
6.2 Dependent Sources
| Type | Description |
|---|---|
| VCVS | Voltage-Controlled Voltage Source: V = μV_control |
| CCVS | Current-Controlled Voltage Source: V = rI_control |
| VCCS | Voltage-Controlled Current Source: I = gV_control |
| CCCS | Current-Controlled Current Source: I = βI_control |
6.3 Source Deactivation Rules
| Source Type | Deactivation Method |
|---|---|
| Voltage Source | Replace with short circuit (0 V) |
| Current Source | Replace with open circuit (0 A) |
| Dependent Source | Leave active (never deactivated) |
7. Voltage and Current Dividers
7.1 Voltage Divider
| Configuration | Formula |
|---|---|
| Series resistors R₁, R₂ | V₁ = V_S × R₁/(R₁ + R₂) |
| General form | V_n = V_S × R_n/R_total |
| Condition | Valid when no current drawn from divider point |
7.2 Current Divider
| Configuration | Formula |
|---|---|
| Parallel resistors R₁, R₂ | I₁ = I_S × R₂/(R₁ + R₂) |
| General form | I_n = I_S × (R_total/R_n) × (1/N) where N resistors in parallel |
| Note | Current flows inversely proportional to resistance |
8. Bridge Circuits
8.1 Wheatstone Bridge
| Parameter | Description |
|---|---|
| Configuration | Four resistors (R₁, R₂, R₃, R₄) in diamond pattern with detector across middle |
| Balance Condition | R₁/R₂ = R₃/R₄ or R₁R₄ = R₂R₃ |
| Balanced Bridge | V_bridge = 0, I_bridge = 0 |
| Application | Precision measurement of unknown resistance |
8.2 Bridge Analysis
- Check balance condition first: if balanced, middle branch can be removed
- If unbalanced, use nodal analysis, mesh analysis, or source transformation
- Delta-Wye transformation useful for complex bridge circuits
9. Delta-Wye Transformations
9.1 Delta to Wye (Δ → Y)
| Wye Resistor | Formula |
|---|---|
| R₁ | R₁ = (R_b R_c)/(R_a + R_b + R_c) |
| R₂ | R₂ = (R_c R_a)/(R_a + R_b + R_c) |
| R₃ | R₃ = (R_a R_b)/(R_a + R_b + R_c) |
| Pattern | Each Y resistor = product of adjacent Δ resistors / sum of all Δ resistors |
9.2 Wye to Delta (Y → Δ)
| Delta Resistor | Formula |
|---|---|
| R_a | R_a = (R₁R₂ + R₂R₃ + R₃R₁)/R₁ |
| R_b | R_b = (R₁R₂ + R₂R₃ + R₃R₁)/R₂ |
| R_c | R_c = (R₁R₂ + R₂R₃ + R₃R₁)/R₃ |
| Pattern | Each Δ resistor = sum of all Y products / opposite Y resistor |
9.3 Special Case: Equal Resistors
| Transformation | Relationship |
|---|---|
| Δ → Y (all R_Δ equal) | R_Y = R_Δ/3 |
| Y → Δ (all R_Y equal) | R_Δ = 3R_Y |
10. Conductance
10.1 Definitions
| Parameter | Description |
|---|---|
| Conductance (G) | G = 1/R, measured in siemens (S) or mho (℧) |
| Ohm's Law | I = GV |
10.2 Conductance Combinations
| Configuration | Formula |
|---|---|
| Series | 1/G_total = 1/G₁ + 1/G₂ + 1/G₃ + ... |
| Parallel | G_total = G₁ + G₂ + G₃ + ... |
- Conductances combine opposite to resistances
- Parallel conductances add directly
11. Resistivity and Resistance
11.1 Material Properties
| Parameter | Formula |
|---|---|
| Resistance | R = ρL/A |
| Variables | ρ = resistivity (Ω·m), L = length (m), A = cross-sectional area (m²) |
| Conductivity | σ = 1/ρ (S/m) |
11.2 Temperature Effects
| Parameter | Formula |
|---|---|
| Temperature Coefficient | R_T = R₀[1 + α(T - T₀)] |
| Variables | α = temperature coefficient (1/°C), T = temperature (°C), R₀ = resistance at T₀ |
| Copper α | 0.00393/°C at 20°C |
| Aluminum α | 0.00429/°C at 20°C |
12. Capacitors in DC Circuits
12.1 Basic Relationships
| Parameter | Formula |
|---|---|
| Capacitance | C = Q/V (farads) |
| Current | i = C(dv/dt) |
| Voltage | v(t) = (1/C)∫i dt + v(0) |
| Energy Stored | W = ½CV² |
12.2 Capacitor Combinations
| Configuration | Formula |
|---|---|
| Series | 1/C_total = 1/C₁ + 1/C₂ + 1/C₃ + ... |
| Series (2 capacitors) | C_total = (C₁C₂)/(C₁ + C₂) |
| Parallel | C_total = C₁ + C₂ + C₃ + ... |
12.3 DC Steady State
- Fully charged capacitor acts as open circuit (I = 0)
- Time constant τ = RC
- Charging: v(t) = V_S(1 - e^(-t/τ))
- Discharging: v(t) = V₀e^(-t/τ)
- Fully charged at t = 5τ (99.3%)
13. Inductors in DC Circuits
13.1 Basic Relationships
| Parameter | Formula |
|---|---|
| Inductance | L = Φ/I (henries) |
| Voltage | v = L(di/dt) |
| Current | i(t) = (1/L)∫v dt + i(0) |
| Energy Stored | W = ½LI² |
13.2 Inductor Combinations
| Configuration | Formula |
|---|---|
| Series | L_total = L₁ + L₂ + L₃ + ... |
| Parallel | 1/L_total = 1/L₁ + 1/L₂ + 1/L₃ + ... |
| Parallel (2 inductors) | L_total = (L₁L₂)/(L₁ + L₂) |
13.3 DC Steady State
- Fully energized inductor acts as short circuit (V = 0)
- Time constant τ = L/R
- Current buildup: i(t) = (V_S/R)(1 - e^(-t/τ))
- Current decay: i(t) = I₀e^(-t/τ)
- Fully energized at t = 5τ (99.3%)
14. First-Order RC Circuits
14.1 Natural Response (No Source)
| Parameter | Formula |
|---|---|
| Time Constant | τ = RC |
| Voltage Decay | v_C(t) = V₀e^(-t/RC) |
| Current | i(t) = (V₀/R)e^(-t/RC) |
14.2 Step Response (DC Source Applied)
| Parameter | Formula |
|---|---|
| Voltage Charging | v_C(t) = V_S + (V₀ - V_S)e^(-t/RC) |
| From Zero | v_C(t) = V_S(1 - e^(-t/RC)) |
| Current | i(t) = (V_S - V₀)/R × e^(-t/RC) |
14.3 Key Time Values
| Time | Percentage of Final Value |
|---|---|
| t = τ | 63.2% |
| t = 2τ | 86.5% |
| t = 3τ | 95.0% |
| t = 4τ | 98.2% |
| t = 5τ | 99.3% |
15. First-Order RL Circuits
15.1 Natural Response (No Source)
| Parameter | Formula |
|---|---|
| Time Constant | τ = L/R |
| Current Decay | i_L(t) = I₀e^(-Rt/L) |
| Voltage | v_L(t) = -RI₀e^(-Rt/L) |
15.2 Step Response (DC Source Applied)
| Parameter | Formula |
|---|---|
| Current Buildup | i_L(t) = I_final + (I₀ - I_final)e^(-Rt/L) |
| From Zero | i_L(t) = (V_S/R)(1 - e^(-Rt/L)) |
| Voltage | v_L(t) = V_Se^(-Rt/L) |
15.3 Initial and Final Conditions
- Inductor current cannot change instantaneously: i_L(0⁺) = i_L(0⁻)
- Inductor voltage can change instantaneously
- At t = 0⁺: inductor acts as open circuit
- At t = ∞: inductor acts as short circuit
16. Circuit Analysis Tips
16.1 Initial Conditions (t = 0⁺)
- Capacitor voltage cannot change instantaneously: v_C(0⁺) = v_C(0⁻)
- Capacitor acts as short circuit at t = 0⁺
- Inductor current cannot change instantaneously: i_L(0⁺) = i_L(0⁻)
- Inductor acts as open circuit at t = 0⁺
16.2 Steady State (t = ∞)
- Capacitor acts as open circuit (fully charged, no current)
- Inductor acts as short circuit (fully energized, no voltage drop)
- All transients have decayed to zero
- Circuit reaches DC steady state
16.3 Problem-Solving Strategy
- Identify circuit type (series, parallel, combination)
- Determine if steady state or transient analysis required
- Find initial conditions if transient (t = 0⁺)
- Find final conditions (t = ∞)
- Calculate time constant (τ = RC or L/R)
- Apply appropriate equation for complete response
- Verify answer using limiting cases (t = 0 and t = ∞)
17. Short Circuit and Open Circuit Analysis
17.1 Short Circuit
| Property | Value |
|---|---|
| Resistance | R = 0 Ω |
| Voltage | V = 0 |
| Current | Can be any value (limited by circuit) |
17.2 Open Circuit
| Property | Value |
|---|---|
| Resistance | R = ∞ Ω |
| Current | I = 0 |
| Voltage | Can be any value (limited by circuit) |
17.3 Applications
- Finding Thévenin voltage: measure open-circuit voltage
- Finding Norton current: measure short-circuit current
- Finding equivalent resistance: short voltage sources, open current sources
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