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Kirchhoff's Laws: Node and Mesh Analysis

  • 3 min
  • Beginner

Las Kirchhoff’s laws are two relationships that express the conservation of charge and energy in a circuit. They allow us to formulate equations when simply recognizing simple resistor combinations is not enough.

Ohm’s law works very well to relate voltage and current in a resistor. What happens if we have a circuit with two sources, three different paths, and five resistors mixed in series and parallel?

In that case, we need to add Kirchhoff’s laws. Within the lumped element model, they are two simple rules that we can apply to very complex networks.

Kirchhoff’s Current Law (KCL)

Imagine a pipe that reaches a “T” junction and splits into two. If 10 liters per minute enter, and 4 liters come out of one side… then 6 liters must necessarily come out of the other. The water does not disappear or get created at the junction.

In electronics, that junction is called a Node.

Kirchhoff’s Current Law (KCL) states:

“The sum of currents entering a node is equal to the sum of currents leaving it.”

Mathematically:

Or equivalently, the algebraic sum is zero:

Practical Example

You look at a schematic and see a point where 3 wires meet:

  • 5 Amperes enter through wire A.
  • 2 Amperes leave through wire B.
  • What happens in wire C?

Kirchhoff’s Voltage Law (KVL)

Suppose you go on an excursion starting from a shelter (0 meters altitude).

  1. You climb a 100m mountain (Gain potential).
  2. You descend into a -50m valley (Lose potential).
  3. You descend another -50m.
  4. You arrive back at the shelter.

If you sum all your ascents and descents along a closed path, the final result is always ZERO. You have returned to the same point.

In electronics, that closed path is called a Mesh or Loop.

Kirchhoff’s Voltage Law (KVL) states:

“In a closed loop, the algebraic sum of all voltages is zero.” Mathematically:

Or equivalently:

Practical Example

You have a 9V battery connected to two resistors in series ( and ).

  • The battery gives you a “rise” of 9V.
  • The current passes through and loses, let’s say, 5V.
  • How much voltage is lost across ?

How to Analyze a Circuit

When you face a circuit and don’t know where to start, follow these steps:

  1. Identify the nodes: group all points connected by ideal conductors and choose a reference node.
  2. Assign currents or voltages: set their directions and polarities consistently. If a solution turns out negative, it means the actual direction is opposite.
  3. Apply KCL or KVL: write independent equations for the necessary nodes or loops.
  4. Apply Ohm’s Law: relate the voltage and current of each resistor using .

In the end, you will obtain a system of equations. Solving it will give you the unknown currents and voltages without relying on whether the circuit can be simplified by sight.

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