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Kirchhoff's Laws, Mesh & Node Analysis

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How to Use

The simulation uses a fixed two-mesh circuit:

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V₁ ──[R₁]── A ──[R₃]── V₂
|            |           |
+           [R₂]         +
−            |           −
└────────────┴───────────┘
            GND

Use the five sliders at the bottom to change V₁, V₂, R₁, R₂, and R₃. All equations and solved values update immediately.

Switch between the three teaching modes using the tabs at the top:

Tab 1 — KVL & KCL

  • Shows current arrows on every branch with magnitudes.
  • KVL Loop 1: \(V_1 - R_1 I_1 - R_2(I_1+I_2) = 0\) — verified to equal zero.
  • KVL Loop 2: \(V_2 - R_3 I_2 - R_2(I_1+I_2) = 0\) — verified to equal zero.
  • KCL at node A: sum of branch currents equals zero.

Tab 2 — Mesh Analysis

  • Each mesh loop is shaded in a different colour with a rotating arrow showing the mesh current direction.
  • Displays the mesh equations in both symbolic and numeric form:
\[ \begin{bmatrix} R_1+R_2 & R_2 \\ R_2 & R_2+R_3 \end{bmatrix} \begin{bmatrix} I_1 \\ I_2 \end{bmatrix} = \begin{bmatrix} V_1 \\ V_2 \end{bmatrix} \]
  • Solved mesh currents and branch currents shown live.

Tab 3 — Node Analysis

  • Node A is highlighted; the reference node (GND) is at the bottom rail.
  • KCL equation at A expanded and solved for \(V_A\):
\[ V_A\!\left(\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}\right) = \frac{V_1}{R_1} + \frac{V_2}{R_3} \]
  • All three branch currents computed from \(V_A\) and verified.

Learning Objectives

After using this simulation, students will be able to:

  1. State KVL and KCL and identify where each applies in a circuit.
  2. Set up mesh-current equations for a two-mesh planar circuit.
  3. Set up a node-voltage equation for a circuit with one unknown node.
  4. Verify that both methods yield the same branch currents.

Key Equations

KVL (sum of voltages around any closed loop = 0):

\[\sum_k V_k = 0\]

KCL (sum of currents leaving any node = 0):

\[\sum_k I_k = 0\]

Mesh equations (matrix form):

[(R_1+R_2)\,I_1 + R_2\,I_2 = V_1] [R_2\,I_1 + (R_2+R_3)\,I_2 = V_2]

Node equation at A:

\[V_A = \frac{V_1/R_1 + V_2/R_3}{1/R_1 + 1/R_2 + 1/R_3}\]