Combined series-parallel networks
Electrical Fundamentals · Lesson 13 · Beginner
Purpose
Reduce series-parallel resistor networks without combining components that do not share the required topology.
Learning objectives
- Mark circuit nodes
- Identify reducible groups
- Work from inner groups outward
- Expand results to branch quantities
Combined networks become manageable when topology is read from nodes rather than from how close components appear on a drawing.
Core theory
Two components are parallel only if both ends connect to the same two nodes. They are series only if their shared node has no other current path.
Reduce one valid group to an equivalent, redraw mentally or on paper, and repeat.
After total current is known, reverse the reductions to recover voltages and branch currents.
| Term | Meaning | Symbol | Unit |
|---|---|---|---|
| Node | Set of conductively connected points at one ideal potential | Not applicable | Not applicable |
| Reduction | Replacement of a subnetwork by an equivalent | Not applicable | Not applicable |
| Topology | Connection relationships independent of drawing shape | Not applicable | Not applicable |
Reduce: series ΣR; parallel (Σ1/R)⁻¹Apply only to groups verified by their nodes.
ΩAssumptions: Ideal source and wires.
- Parallel group: 6 Ω || 3 Ω = 2 Ω.
- Total: Req = 4 + 2 = 6 Ω; It = 12/6 = 2 A.
- Expand: Drop on 4 Ω is 8 V, leaving 4 V across the parallel pair; branches are 0.667 A and 1.333 A.
Req = 6 Ω and source current = 2 A.
Reasonableness check: Branch currents total 2 A and voltage drops total 12 V.
- Combining components based on drawing position
- Failing to redraw after reduction
- Rounding intermediate values too early
Where this appears in practice
Reduction supports sensor networks, load banks, bias circuits, and preliminary fault reasoning.
Knowledge check
What proves two resistors are parallel?
Both terminals of each connect to the same two nodes. Visual alignment alone is insufficient.
Answer: Both terminals of each connect to the same two nodes. Visual alignment alone is insufficient.
Practical exercise
Solve 10 Ω in series with 20 Ω || 30 Ω on 22 V and verify both conservation checks.
Summary
- Read nodes, not drawing proximity
- Reduce one valid group at a time
- Reverse the reduction to recover branch values
Sources and review
- The International System of Units (SI Brochure): BIPM; 9th edition, updated 2026; International
- SI Units: Electric Current: NIST; Current web edition; United States / SI reference
- University Physics Volume 2: OpenStax; Current web edition; Physics reference
Editorial review date: 2026-08-21. Professional electrical review is pending.