How to Solve Electrical Circuits with Kirchhoff's Current Law (KCL)

Analyzing electric circuits often goes beyond simple Ohm’s Law and basic series-parallel simplifications. In such cases, Kirchhoff’s Laws, developed by the German physicist Gustav Kirchhoff, offer a powerful framework for analyzing more intricate circuits. Although Kirchhoff introduced two fundamental laws (KCL and KVL), here we’ll focus on Kirchhoff’s Current Law (KCL) and nodal analysis.

Introduction to Kirchhoff’s Current Law (KCL)

KCL is based on the principle of charge conservation—what goes into a node must come out. In other words, at any node (a junction point between circuit elements), the sum of currents flowing into the node must equal the sum of currents flowing out.

Steps to Solve Circuits Using KCL

Let's break down the steps to use KCL for circuit analysis, using sample problems to clarify.

Step 1: Review the Circuit Setup

Start by understanding the circuit layout. Identify the values of resistances and sources provided, label unknowns, and note how many nodes (junctions) you’ll need to analyze. For example, in a circuit with three resistors and two current sources, you may only need to find the voltages at two nodes and the current through one branch.

Step 2: Choose a Reference Ground

Designate one node as the ground (reference point) to simplify calculations. Typically, the bottom or a central node is chosen, but any node will work.

Step 3: Define Current Directions and Write Branch Equations

Assign current directions for each branch. The actual current may differ, but if it flows opposite to the assumed direction, it will appear as a negative value in your solution.

For each branch, express current as a function of node voltages and resistance. For example, if you assume the current through a resistor R₃ flows downwards from node N₁ to N₂, the current I₃ would be:
I₃ = (V_N1 − V_N2) / R₃

Step 4: Apply KCL to Each Node

Write an equation for each node by summing currents entering and leaving the node, setting the total equal to zero. For example, if currents I₁ and I₂ enter node N₁ and I₃ exits, the KCL equation for N₁ would be:
I₁ + I₂ − I₃ = 0

Step 5: Solve the System of Equations

If you have one unknown, solve directly. For multiple unknowns, set up simultaneous equations and solve using substitution or matrix methods.

Example 1: Circuit with Current Source

Consider a circuit with three resistors (none in series or parallel) and two current sources, as shown in a figure.

• Identify known values: Given two current sources, known resistor values, and two nodes N₁ and N₂.

• Choose ground: Set N₂ as the reference ground.

• Write branch equations: For example, if the current through R₃ is unknown, represent it as I₃ = (V_N1 − V_N2) / R₃.
• Apply KCL: At N₁, the equation becomes I₁ + I₂ = I₃.
• Solve: Substitute known values for I₁ and I₂, and solve for I₃.

Example 2: Circuit with Voltage Source

Now, let's consider a circuit with a voltage source instead of a current source, with three resistors where R₂ and R₃ are in parallel and in series with R₁.

• Review the circuit: Note that R₂ and R₃ are parallel resistors.
• Choose ground: Ground is already defined in this example.
• Define branch currents: Define each current in terms of the unknown voltage V₁ at the node.
• Set up equations: The currents through R₂ and R₃ add up to the current through R₁, represented by I₁ = I₂ + I₃.
• Solve: Use algebra or matrix methods to find V₁ and corresponding currents.

Summary of Steps:

1. Review circuit elements and identify all known and unknown values.
2. Select a reference ground.
3. Define current directions for each branch.
4. Apply KCL to write equations for each node.
5. Solve the system of equations, checking for reasonableness in your final values.

Using these steps, KCL becomes an effective tool for analyzing complex circuits with multiple branches, resistors, and sources.