In electrical network theory, correctly identifying whether a circuit is planar or non-planar decides which analysis technique (mesh or nodal) is most effective. This post explains both types, gives clear examples, lists applicable theorems, and shows which analysis method to use — useful for SSC JE, RRB JE, GATE and other competitive exams.
Table of contents
- What is a Planar Circuit?
- What is a Non-Planar Circuit?
- Graph-theory condition (K₅ and K₃,3)
- Theorems & Analysis Methods (which apply where)
- Worked examples (one planar, one non-planar)
- Key takeaways
- FAQs
1. What is a Planar Circuit?
A planar circuit is a circuit that can be drawn on a plane without any of its branches (wires) crossing one another except at common nodes. If you can rearrange the drawing so that no two branches overlap, the circuit is planar.
Planar examples
- Simple RLC ladder networks arranged in series/parallel
- Standard Wheatstone bridge (the usual 4-resistor bridge drawn without extra crossings)
- Square or triangular meshes with elements on edges
2. What is a Non-Planar Circuit?
A non-planar circuit cannot be drawn on the plane without at least one crossing of branches. No matter how you redraw the network, an edge crossing remains.
Non-planar examples
- Any circuit whose graph contains a K₅ (complete graph of 5 vertices) or a K₃,3 (complete bipartite 3-by-3) as a subgraph is non-planar.
- Wheatstone bridge with an extra diagonal connection that creates unavoidable crossings
3. Graph-theory condition (K₅ and K₃,3)
Kuratowski's theorem states: a finite graph is planar if it does not contain a subgraph that is a subdivision of K₅ or K₃,3. In plain language: if you can reduce a circuit's graph to K₅ or K₃,3 by removing or contracting edges, the graph is non-planar.
4. Theorems & Analysis Methods — Which Apply Where?
For Planar Circuits
- Mesh (Loop) Analysis — applicable and often easiest. Define independent mesh currents and apply KVL around each mesh.
- KVL and Mesh analysis simplify many planar networks.
- Standard theorems: Thevenin, Norton, Superposition, Maximum Power Transfer all apply.
For Non-Planar Circuits
- Mesh analysis is usually not applicable (or becomes awkward) because you cannot define independent non-overlapping meshes due to crossings.
- Nodal Analysis (KCL) is the go-to method — it works for any circuit whether planar or not.
- Graph-based methods like cut-set and tie-set analysis (from network graph theory) are useful.
- Thevenin and Norton equivalents are still valid.
5. Worked Examples
Example A — Planar circuit (Mesh analysis)
Circuit: A square loop with four resistors R1, R2, R3, R4 and a voltage source V in one branch. Drawn so no wires cross.
Approach: Use two or three mesh currents (depending on how you split the loop). Apply KVL to each independent mesh and solve simultaneous equations for mesh currents. Convert mesh currents to branch currents where needed.
Why this works: Mesh currents are independent because the graph is planar and meshes do not overlap in a crossing manner.
Example B — Non-planar circuit (Nodal analysis)
Circuit: A Wheatstone bridge with an extra diagonal resistor or connection that forces an unavoidable crossing.
Approach: Choose a reference node. Write KCL at each non-reference node (N-1 equations for N nodes). Solve the node-voltage equations (use matrix methods or substitution). From node voltages obtain branch currents and powers.
Why mesh fails: Because mesh loops would overlap or share edges in a way that makes independent mesh current definitions impossible or inconsistent.
6. Key takeaways
- Identify planarity first — it tells you whether mesh analysis is practical.
- Nodal analysis (KCL) is universal and always works.
- Many classical theorems (Thevenin, Norton, Superposition) apply to both planar and non-planar circuits.
- Graph theory (Kuratowski's theorem) gives a formal test for planarity.
7. FAQs (Quick)
Q: Can Thevenin's theorem be used on non-planar circuits?
A: Yes — Thevenin and Norton are topology-independent.
Q: Why is mesh analysis simpler for planar circuits?
A: Because you can define independent non-overlapping meshes and apply KVL directly, reducing the number of equations in many cases.
Q: If a circuit is non-planar, is nodal analysis always easier?
A: Generally yes — nodal/KCL avoids problems caused by crossing branches. However the number of nodes vs meshes matters; choose the method with fewer equations.