Fundamental circuit analysis
What are Kirchhoff's laws?
Kirchhoff's laws are two fundamental rules used to analyze electrical circuits. They describe how electric current behaves at a junction and how voltage behaves around a closed loop.
Kirchhoff's Current Law is based on conservation of electric charge. Kirchhoff's Voltage Law is based on conservation of energy.
Together, the two laws make it possible to create simultaneous equations for circuits containing several branches, nodes, voltage sources, and resistive elements.
Junction-current relationship
Kirchhoff's Current Law formula
Kirchhoff's Current Law, commonly abbreviated as KCL, states that the algebraic sum of all currents meeting at a circuit junction is zero.
Current-law equationΣI = 0
Equivalent statementTotal current entering = total current leaving
One consistent sign convention is to treat currents entering the junction as positive and currents leaving the junction as negative. The reverse convention is also valid when applied consistently.
Closed-loop relationship
Kirchhoff's Voltage Law formula
Kirchhoff's Voltage Law, commonly abbreviated as KVL, states that the algebraic sum of all voltage changes around a complete closed loop is zero.
Voltage-law equationΣV = 0
Equivalent statementTotal voltage rises = total voltage drops
A common convention treats voltage rises as positive and voltage drops as negative. Reversing every sign also produces a valid equation.
Automatic unknown-term calculation
How the Kirchhoff's law calculator works
Select either KCL or KVL, choose which term is unknown, and enter all remaining signed values. The calculator adds the known terms and changes the sign of their total.
Unknown signed valuex = −Σ known values
The resulting unknown causes the complete algebraic sum to equal zero, satisfying the selected Kirchhoff law.
The result panel also displays the formula, numerical substitution, known-value sum, and final verification.
Worked junction example
Kirchhoff's Current Law example
Suppose a junction has a signed current of 5 A, another signed current of −2 A, and one unknown branch current.
Starting equation5 + (−2) + I3 = 0
Rearranged equationI3 = −(5 − 2)
ResultI3 = −3 A
The negative result indicates that the unknown current flows in the direction represented by the negative sign convention.
Worked loop example
Kirchhoff's Voltage Law example
Consider a loop containing a 12 V rise, a −4 V drop, a −3 V drop, and one unknown voltage.
Starting equation12 + (−4) + (−3) + V4 = 0
Rearranged equationV4 = −(12 − 4 − 3)
ResultV4 = −5 V
The unknown is a voltage drop of 5 volts under the selected sign convention.
Direction and polarity
How to choose current and voltage signs
Correct signs are essential because Kirchhoff equations are algebraic rather than simple unsigned totals.
- For KCL, define currents entering a node as positive and currents leaving as negative, or use the reverse convention.
- For KVL, choose a direction in which to move around the circuit loop.
- Record a voltage rise when moving from lower potential to higher potential.
- Record a voltage drop when moving from higher potential to lower potential.
- Never change the sign convention halfway through the equation.
Circuit terminology
Nodes, branches, and loops
- A node is a connection point shared by two or more circuit elements.
- A branch is a path between two nodes containing one or more circuit elements.
- A loop is any closed path through a circuit.
- A mesh is a loop that does not contain another loop inside it.
KCL is normally applied at nodes, while KVL is normally applied around loops or meshes.
Larger circuit systems
Using Kirchhoff's laws for multiple unknowns
A circuit containing several unknown currents or voltages normally requires more than one independent equation. KCL equations are written for selected nodes, and KVL equations are written for selected loops.
Ohm's law relationships such as V = IR are then substituted into the Kirchhoff equations. The resulting simultaneous equations can be solved using algebra, elimination, substitution, or matrix methods.
This calculator solves one unknown signed term at a time. It does not currently solve a full system containing several unknown branch currents or loop voltages.
Practical uses
Where Kirchhoff's laws are used
- DC resistor-network analysis.
- Parallel branch-current calculations.
- Multi-loop circuit analysis.
- Laboratory circuit verification.
- Electronics troubleshooting.
- Mesh-current and nodal-voltage methods.
- Power-system and network modeling.
Calculation accuracy
Common Kirchhoff calculation mistakes
- Treating all currents or voltages as positive values.
- Changing the assumed current direction during a calculation.
- Reversing battery or component polarity.
- Omitting one branch current from a node equation.
- Omitting one voltage rise or drop from a loop equation.
- Mixing amperes and milliamperes without conversion.
- Mixing volts and millivolts without conversion.
Tool scope
Calculator assumptions and limitations
This calculator applies the algebraic zero-sum form of Kirchhoff's Current Law and Kirchhoff's Voltage Law. It supports real, finite, signed numerical values.
It does not automatically construct a circuit model, detect independent nodes or loops, calculate resistance-based voltage drops, or solve simultaneous systems with several unknowns.
AC circuits may require complex impedance, phase angle, and phasor calculations, which are outside this calculator's current scope.
Related electrical tools
Continue analyzing the circuit
Use the Ohm's Law Calculator to calculate voltage, current, resistance, or electrical power.
Use the Current Divider Calculator to calculate current distribution between two parallel resistor branches.
Use the Voltage Divider Calculator to calculate voltage distribution across two series resistors.
Use the Series and Parallel Resistance Calculator to calculate equivalent resistance in resistor networks.
Use the AC Impedance Calculator for circuits containing resistance, inductive reactance, or capacitive reactance.
Common questions
Kirchhoff's law calculator FAQ
What does Kirchhoff's Current Law state?
Kirchhoff's Current Law states that the algebraic sum of currents at a junction is zero. Equivalently, the total current entering a junction equals the total current leaving it.
What does Kirchhoff's Voltage Law state?
Kirchhoff's Voltage Law states that the algebraic sum of all voltage rises and voltage drops around a closed circuit loop is zero.
How should positive and negative signs be entered?
For KCL, assign one sign to currents entering a junction and the opposite sign to currents leaving it. For KVL, assign one sign to voltage rises and the opposite sign to voltage drops. The chosen convention must remain consistent.
Can the calculator solve more than three circuit terms?
Yes. The calculator supports between two and six signed current or voltage terms and can calculate any one selected unknown term.
Does the order of circuit values matter?
The arithmetic result depends on the signs and magnitudes rather than the displayed order. However, keeping values in circuit-traversal order makes KVL calculations easier to understand and verify.
Can Kirchhoff's laws be used with AC circuits?
Yes, but AC circuit analysis may require complex impedances and phasor quantities. This calculator currently accepts real signed numerical values and does not perform complex-number calculations.