Planar and Non-Planar Circuits: Analysis and Applications

A planar circuit is a circuit that can be drawn on a flat surface without any wires crossing each other.

A non-planar circuit is a circuit that cannot be drawn on a flat surface without wires crossing. The difference decides which analysis tools apply: mesh analysis needs a planar layout, while other methods handle both cases. Graph theory and the loop current method, described below, cover the two families.

What is Graph Theory?

Graph theory is the branch of mathematics that studies the properties and relationships of graphs. A graph consists of nodes (vertices) and edges (branches) connecting those nodes, and it models phenomena across science, engineering and the social sciences.

One of the applications of graph theory is to represent electrical circuits. Each element in a circuit (such as a resistor, a capacitor, or a voltage source) can be represented by an edge in a graph. In a graph, each node represents a junction point or terminal in a circuit. The current flow direction in a circuit is shown by arrows on the edges, forming an oriented graph.

What is a Planar Circuit?

A planar circuit is a circuit that can be drawn on a flat surface without any wires crossing each other. Equivalently, a planar circuit is a circuit whose oriented graph can be embedded on a plane without any edges crossing each other. A planar circuit has some advantages over a non-planar circuit, such as:

• It is easier to visualize and draw.
• Its meshes map one-to-one onto mesh-analysis equations, keeping the algebra tidy.
• It can be analyzed using mesh analysis or nodal analysis, which are systematic methods based on Kirchhoff’s laws.

What is a Non-Planar Circuit?

A non-planar circuit is a circuit that cannot be drawn on a flat surface without any wires crossing each other.

Equivalently, a non-planar circuit is one whose oriented graph cannot be embedded on a plane without edges crossing. Compared with a planar design, a non-planar one brings some drawbacks:

• It is harder to visualize and draw.
• Its crossings hide the loop structure, so setting up equations takes more care.
• It rules out mesh analysis, which applies only to planar circuits, though nodal analysis still works.

How to Analyze Planar and Non-Planar Circuits?

The loop current method, built on Kirchhoff’s voltage law (KVL), handles planar and non-planar circuits alike. Its steps:

1. Identify the loops in the circuit. A loop is any closed path; a mesh is the special case of a loop enclosing no other loop. Where a current source sits between two meshes, the pair merges into a supermesh for the analysis.
2. Assign loop currents to each loop. A loop current is an imaginary current that flows around the loop in either a clockwise or counterclockwise direction. The direction of the loop current can be chosen arbitrarily, but it must be consistent throughout the analysis.
3. Write KVL equations for each loop. A KVL equation states that the algebraic sum of the voltages around any closed loop is zero. The voltage across an element depends on its type and polarity, as well as the direction of the loop current relative to the element current.
4. Solve the system of equations for the unknown loop currents by substitution, elimination, matrix inversion or Cramer’s rule.
5. Find the element currents and voltages using the loop currents. The element current is equal to the sum or difference of the loop currents passing through it, depending on their directions. The element voltage can be found using Ohm’s law or other relations for different types of elements.

How to Identify Planar and Non-Planar Circuits?

Two quick tests decide whether a circuit is planar:

• If the circuit can be redrawn without any wires crossing each other, then it is planar.
• If the circuit cannot be redrawn without any wires crossing each other, then it is non-planar.

Sometimes, a circuit may appear non-planar initially but can be redrawn as planar by rearranging elements or nodes. For example, consider a circuit where two resistors cross each other. At first glance, it seems non-planar.

Moving one resistor to a different position, however, yields an equivalent planar circuit.

Therefore, this circuit is actually planar.

What are Some Applications of Planar and Non-Planar Circuits?

Both circuit families appear across electrical engineering:

• Planar circuits are widely used in printed circuit boards (PCBs), which are thin sheets of insulating material with conductive tracks printed or etched on them. PCBs are used to connect electronic components in devices such as computers, smartphones, radios, etc.
• Non-planar circuits are often used in three-dimensional (3D) integrated circuits (ICs), which are stacks of multiple layers of electronic components connected by vertical wires or vias.



• 3D ICs gain density, power efficiency, speed and performance over conventional 2D ICs.
• Planar and non-planar circuits can also be used to model various physical phenomena that involve networks or graphs, such as fluid flow, heat transfer, traffic flow, social networks, etc.

Summary

Planarity decides which analysis shortcuts a circuit allows. Graph theory supplies the language, and the loop current method supplies the equations.

The main points to remember:

• A planar circuit is a circuit that can be drawn on a flat surface without any wires crossing each other.
• A non-planar circuit is a circuit that cannot be drawn on a flat surface without any wires crossing each other.
• Graph theory is a branch of mathematics that studies the properties and relationships of graphs.
• An oriented graph is a graph with arrows on its edges indicating the direction of current flow in an electrical circuit.
• The loop current method is based on Kirchhoff’s voltage law (KVL) and involves writing equations for all the loops in a circuit and solving them for the unknown loop currents.
• Mesh analysis applies only to planar circuits; nodal analysis works on either kind.
• Non-planar circuits still yield to the loop current method with general loops, or to nodal analysis.
• Planar circuits are widely used in printed circuit boards (PCBs), while non-planar circuits are often used in three-dimensional (3D) integrated circuits (ICs).