Kirchhoff's Laws

Statement of laws

KCL (Current Law): The algebraic sum of currents at any node is zero.

Σ I_in = Σ I_out  (at every node)

KVL (Voltage Law): The algebraic sum of voltage drops around any closed loop is zero.

Σ V_drops = 0  (around any closed loop)

How to apply

1. Identify nodes (junctions where 3+ wires meet) and loops (closed paths).
2. Assign current direction to each wire (arbitrary; sign of answer tells you actual direction).
3. Apply KCL at each node — Σ I = 0.
4. Apply KVL around each independent loop — Σ V_drops = 0.
5. Use Ohm's law (V = IR) to relate V and I at each resistor.
6. Solve the system of equations.

Example: two-loop circuit

Battery V₁ in loop 1, battery V₂ in loop 2, resistors R₁, R₂, R₃ shared.

Mesh analysis with loop currents I₁ and I₂ (each loop's "circulation"):

Loop 1: V₁ = I₁·R₁ + (I₁ - I₂)·R₃
Loop 2: V₂ = I₂·R₂ + (I₂ - I₁)·R₃

Two equations, two unknowns (I₁, I₂). Solve via substitution or matrix.

JavaScript — circuit analysis with Kirchhoff

// Solve simple two-loop circuit using mesh analysis
function twoLoopCircuit(V1, V2, R1, R2, R3) {
  // Mesh equations:
  // V1 = I1·(R1 + R3) − I2·R3
  // V2 = -I1·R3 + I2·(R2 + R3)
  // Or: V1 = I1·R1 + (I1-I2)·R3
  //     V2 = I2·R2 + (I2-I1)·R3
  const A = [
    [R1 + R3, -R3],
    [-R3, R2 + R3]
  ];
  const b = [V1, V2];
  // Solve 2x2 linear system
  const det = A[0][0]*A[1][1] - A[0][1]*A[1][0];
  const I1 = (b[0]*A[1][1] - b[1]*A[0][1]) / det;
  const I2 = (A[0][0]*b[1] - A[1][0]*b[0]) / det;
  return [I1, I2];
}

// Example: 6V battery, 3V battery, R1=R2=R3=1Ω
const [I1, I2] = twoLoopCircuit(6, 3, 1, 1, 1);
console.log(`I1 = ${I1.toFixed(2)} A, I2 = ${I2.toFixed(2)} A`);

// Verify KVL
function verifyKVL(V1, I1, R1, I2, R3) {
  // Loop 1: V1 - I1·R1 - (I1-I2)·R3 should = 0
  return V1 - I1*R1 - (I1-I2)*R3;
}

console.log(`KVL check: ${verifyKVL(6, I1, 1, I2, 1).toExponential(2)}`); // ~0

// Voltage divider: simplest KVL
function voltageDividerKVL(V_in, R1, R2) {
  // V_in = I·R1 + I·R2 → I = V_in / (R1+R2)
  // V_out = I·R2
  const I = V_in / (R1 + R2);
  return { current: I, V_R1: I*R1, V_R2: I*R2 };
}

console.log(voltageDividerKVL(9, 100, 200));

// Current divider: simplest KCL
function currentDividerKCL(I_total, R1, R2) {
  // Both branches at same V; I = V/R; total V = I·(R1·R2/(R1+R2))
  // Currents: I1·R1 = I2·R2 = V (both branches have same V)
  const V = I_total * R1 * R2 / (R1 + R2);
  return { I_R1: V/R1, I_R2: V/R2, V_branch: V };
}

console.log(currentDividerKCL(2, 100, 200));

// Bridge circuit balance condition
function isBridgeBalanced(R1, R2, R3, R4) {
  // Wheatstone bridge balanced when R1/R2 = R3/R4 → no current in galvanometer
  return Math.abs(R1*R4 - R2*R3) < 1e-9;
}

console.log(isBridgeBalanced(100, 200, 1000, 2000));  // true (1:2 ratios match)
console.log(isBridgeBalanced(100, 200, 1000, 2100));  // false

// Simple node analysis: 3 nodes, 5 branches example
function nodalAnalysisExample(V_source, R1, R2, R3) {
  // Star network: ground at one node, V_source connected to another
  // Various paths
  // For specific example, write equations and solve
  // (Skipping general implementation — too complex for snippet)
  return "Use SPICE for general circuits";
}

Where Kirchhoff's laws matter

• Circuit design. Foundation of all analog and digital circuit analysis.
• SPICE simulation. Industry-standard tool — solves Kirchhoff's equations numerically.
• Power grid analysis. Massive application; load flow studies use generalized Kirchhoff.
• Operational amplifiers. Op-amp circuits analyzed via KCL/KVL with assumption of "virtual short" between inputs.
• Bridge circuits. Wheatstone bridge for sensitive resistance measurement.
• Education. Bedrock of EE curriculum; first lawyers learn after Ohm.
• Filters and amplifiers. All audio, RF, signal-processing circuits use KVL/KCL methods.

Common mistakes

• Sign errors. Trace direction matters. Be consistent — voltage rise (− to +) is positive going one way; drop the opposite. Currents arbitrary direction; sign tells you actual.
• Missing node or loop. Identify all nodes and select independent loops. Don't double-count or miss any.
• Confusing series and parallel. Series — same current. Parallel — same voltage. Mistakes break KCL/KVL.
• Forgetting capacitor and inductor V-I relations. For DC steady state, capacitor open (I=0), inductor short (V=0). For AC, use impedances.
• Ignoring polarity of sources. Battery's + side is higher V than − side. KVL must account for this consistently.
• Treating reciprocal-resistor problems as additive. Resistors in series add; in parallel, reciprocals add. Capacitors are opposite. Don't mix up.