Kirchhoff Circuit Laws

Solve the fixed two-mesh DC resistor circuit with clockwise mesh currents I1,I2 sharing resistor R3.

Kirchhoff Circuit Laws

Concept Overview

Kirchhoff's Circuit Laws are fundamental principles used to analyze complex electrical circuits. Kirchhoff's Current Law (KCL) states that the sum of currents entering a junction must equal the sum of currents leaving it (conservation of charge). Kirchhoff's Voltage Law (KVL) states that the directed sum of the electrical potential differences (voltage) around any closed network is zero (conservation of energy). In this module, we use mesh analysis to apply KVL directly, generating a system of linear equations to solve for the unknown currents.

Mathematical Definition

For the two-mesh circuit in this simulation, we define two clockwise mesh currents, I₁ and I₂. The left source is positive above and the right source is positive below. Both clockwise traversals therefore cross their own source from negative to positive. Positive shared current I1 − I2 goes downward. By applying Kirchhoff's Voltage Law (KVL) to both loops, taking into account the shared resistor R₃, we derive the following linear equations:

Loop 1: (R₁ + R₃)I₁ - R₃I₂ = E₁\n Loop 2: -R₃I₁ + (R₂ + R₃)I₂ = E₂

This system can be solved using Cramer's rule. The primary determinant of the system must be positive for a valid purely resistive circuit:

Δ = R₁R₂ + R₁R₃ + R₂R₃ > 0

Worked Example

Let's manually calculate the currents for a specific scenario using the controls provided in the module. By substituting the values into our KVL equations, we can trace the mathematical steps required to solve for the unknown mesh currents and the shared branch current.

Set Left source E1 to 6 V.
Set Right source E2 to 0 V.
Set Left resistance R1, Right resistance R2, and Shared resistance R3 to 2.

Substituting these values into our KVL equations:

(2 + 2)I₁ - 2I₂ = 6 => 4I₁ - 2I₂ = 6
-2I₁ + (2 + 2)I₂ = 0 => -2I₁ + 4I₂ = 0

From the second equation, we find 2I₁ = 4I₂, which simplifies to I₁ = 2I₂. Substituting this back into the first equation:

4(2I₂) - 2I₂ = 6
8I₂ - 2I₂ = 6
6I₂ = 6 => I₂ = 1 A

Since I₁ = 2I₂, we find I₁ = 2 A. The shared branch current is I₁ - I₂ = 2 - 1 = 1 A. The power supplied by the sources is (6)(2) + (0)(1) = 12 W. The power dissipated by the resistors is (2)(2²) + (2)(1²) + (2)(1²) = 8 + 2 + 2 = 12 W, confirming energy conservation.

Experiment

Use the interactive controls in the Kirchhoff Circuit Laws module to observe how the mesh currents shift. These examples demonstrate the superposition principle and current reversal based on relative source strengths.

Set Left source E1 to 6 and Right source E2 to 6, with all resistors at 2 Ω. Observe that I₁ and I₂ both become 3 A. The shared branch current cancels out to exactly 0 A.
Set Left source E1 to 0 and Right source E2 to 6. Notice that the shared branch current reverses direction (becomes -1 A), which is indicated by the arrow in the schematic reversing its orientation.
Set Left source E1 and Right source E2 both to 0. Every current, supplied power and dissipated power is 0. The shared-current arrow disappears rather than claiming a direction for zero.
Set Left resistance R1 to 0. The genuine visible input error replaces computed values; ideal shorts are outside this model. Restore R1 to 1, then set Right resistance R2 and Shared resistance R3 to 1 and both sources to 12. Both mesh currents are 12 A, shared current is 0 and both powers are 288 W.
Clear Right source E2 and observe the accessible raw-input error. Restore 0; the circuit and physical results recover. A residual is a computed floating-point check, not a forced zero label.

Limitations

This educational simulation is designed to clearly illustrate the principles of Kirchhoff's Laws, but it has several constraints compared to a full circuit solver.

Fixed Topology: This module strictly simulates a specific two-mesh circuit with a shared central branch. It is not an arbitrary circuit editor.
DC Only: The simulation only models steady-state Direct Current (DC) with purely resistive loads. It does not account for transient behaviors, AC sources, capacitors, or inductors.
Ideal Components: Sources are treated as ideal voltage sources with zero internal resistance. The connecting wires are assumed to have zero resistance. Source polarities are fixed as shown and input voltages are nonnegative. The SVG shows topology and reference directions, not physical wire lengths. Displayed numbers use eleven significant digits; tiny loop residuals can remain from floating-point arithmetic and are not silently snapped to zero.

References

For a more comprehensive understanding of circuit analysis and Kirchhoff's Rules, please refer to the following authoritative resources.

University Physics Volume 2: 10.3 Kirchhoff's Rules (OpenStax)

Kirchhoff Circuit Laws