Series-Parallel Circuits
Learning Objectives
- Identify series-parallel combinations in complex circuits
- Apply systematic reduction techniques for equivalent resistance
- Analyze circuits with both series and parallel elements
- Use step-by-step simplification strategies
What are Series-Parallel Circuits?
Most real-world circuits contain both series and parallel combinations. These circuits require systematic analysis techniques to simplify and solve.
Series-Parallel Circuit Example
R₁ and R₄ in series with a parallel combination of R₂ and R₃
Systematic Analysis Strategy
💡 Step-by-Step Approach
- Identify combinations: Find all series and parallel groupings
- Start from the far end: Simplify the circuit from the load toward the source
- Replace components: Calculate equivalent resistance for each group
- Repeat: Continue until you have a single equivalent resistance
- Work backward: Find voltages and currents in individual components
Step 1: Identify Parallel Combination
R₂ and R₃ are in parallel. Calculate their equivalent:
Parallel R₂ || R₃
R_parallel = (R₂ × R₃) / (R₂ + R₃)
Step 2: Combine with Series Resistors
Now the parallel combination is in series with R₁ and R₄:
Total Resistance
R_total = R₁ + R_parallel + R₄
Series-Parallel Circuit Calculator
Circuit Parameters
Results
R_parallel (R₂||R₃)
15 Ω
R_total
35 Ω
Total Current
0.69 A
V across R₂||R₃
10.3 V
I through R₂
0.34 A
I through R₃
0.34 A
Practice Problem
Calculate Total Resistance and Current
Given: V = 12V, R₁ = 100Ω, R₂ = 100Ω (parallel), R₃ = 100Ω (parallel), R₄ = 100Ω
Show Solution
Step 1: Parallel combination of R₂ and R₃
R_parallel = (100 × 100) / (100 + 100)
R_parallel = 10000 / 200 = 50ΩStep 2: Total resistance
R_total = R₁ + R_parallel + R₄
R_total = 100 + 50 + 100 = 250ΩStep 3: Total current
I_total = V / R_total
I_total = 12V / 250Ω = 0.048A = 48mAAnswer: R_total = 250Ω, I_total = 48mA
Summary
- Series-Parallel: Combines both series and parallel elements
- Strategy: Simplify from far end toward source
- Parallel first: Often easier to combine parallel groups first
- Work backward: Use equivalent circuits to find branch values
- KVL/KCL: Still apply to any simplified section