Kirchhoff Laws

Overview

Kirchhoff's laws are fundamental rules for analyzing electrical circuits. The current law (KCL) applies to nodes, and the voltage law (KVL) applies to loops.

Key Formulas

Kirchhoff's Current Law (KCL)

$\sum I_{in} = \sum I_{out}$

Total current entering a node equals total current leaving.

Kirchhoff's Voltage Law (KVL)

$\sum V = 0$

Sum of voltage drops around any closed loop is zero.

Alternative Names

Kirchhoff's laws are known by several alternative names:

Kirchhoff's Current Law (KCL):

• Kirchhoff's first law
• Kirchhoff's point rule
• Kirchhoff's junction rule
• Kirchhoff's nodal rule
• Kirchhoff's first rule

Kirchhoff's Voltage Law (KVL):

• Kirchhoff's second law
• Kirchhoff's loop rule
• Kirchhoff's mesh rule
• Kirchhoff's second rule

Kirchhoff's Current Law Example

Given the following currents in a circuit junction:

• $I_1 = 0.1 \text{ A}$ (toward junction)
• $I_2 = 0.2 \text{ A}$ (toward junction)
• $I_3 = 0.1 \text{ A}$ (toward junction)
• $I_4 = -0.05 \text{ A}$ (away from junction)
• $I_5 = -0.15 \text{ A}$ (away from junction)

Apply Kirchhoff's Current Law to find $I_6$:

$\sum I = I_1 + I_2 + I_3 + I_4 + I_5 + I_6 = 0$

$(0.1 \text{ A}) + (0.2 \text{ A}) + (0.1 \text{ A}) + (-0.05 \text{ A}) + (-0.15 \text{ A}) + I_6 = 0$

$0.2 \text{ A} + I_6 = 0$

$I_6 = -0.2 \text{ A}$

The negative sign indicates that $I_6$ flows away from the junction.

Voltage Sign Convention

When applying Kirchhoff's Voltage Law (KVL), the sign convention depends on the direction of traversal:

• Voltage drops are positive when measured in the direction of assumed current flow
• Voltage rises are positive when traversing from negative to positive terminal of a source
• The sum of all signed voltages around any closed loop equals zero: $\sum U = 0$

The choice of clockwise or anticlockwise traversal is arbitrary, but must be applied consistently throughout the analysis.

References

• Original Source