Kirchhoff's Laws: KCL and KVL Explained with Examples
Have you ever wondered how to calculate current in different branches of an electrical circuit or find the voltage across a resistor when several components are connected together? Kirchhoff's laws help us solve these problems.
There are two laws: Kirchhoff's Current Law (KCL), which deals with current at a junction, and Kirchhoff's Voltage Law (KVL), which deals with voltage around a closed loop. Both are important tools for analysing electrical circuits.
Interactive Kirchhoff Circuit Lab
Change each source voltage or resistance. The diagram and calculated values update automatically. The simulator uses ideal voltage sources and resistors connected between a common reference rail and one shared node.
KCL mode: the algebraic sum of branch currents at the shared node is zero. Positive current is defined from each source branch toward the shared node.
What Are Kirchhoff's Laws?
Kirchhoff's laws are two basic rules used to understand and calculate electrical quantities in a circuit. They are named after the physicist Gustav Kirchhoff.
The idea behind these rules is straightforward: electric charge does not simply disappear at a junction, and energy is conserved as we travel around a closed circuit loop.
First law: KCL
Current at a junction: the total current entering a junction equals the total current leaving it.
Second law: KVL
Voltage around a loop: the algebraic sum of voltage changes around a closed loop is zero.
1. Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law is also called the junction rule or current law. A junction is a point in a circuit where three or more conducting branches meet.
Whenever current reaches a junction, it divides among the available outgoing branches or combines with other currents. The amount entering must equal the amount leaving, provided there is no net accumulation of charge at that junction.
Another mathematical form is:
In the second equation, choose one current direction as positive and the opposite direction as negative. For example, currents entering the junction can be positive and currents leaving can be negative.
Understand KCL using a junction diagram
KCL Solved Numerical: Find the Unknown Current
Question: A junction receives 7 A of current. Two branches carry 2 A and 1.5 A away from the junction. Find the current in the third outgoing branch.
Step 1: Write the KCL equation.
Step 2: Substitute the known values.
Step 3: Rearrange the equation.
Final answer:
2. Kirchhoff's Voltage Law (KVL)
Kirchhoff's Voltage Law states that the algebraic sum of all voltage changes around any closed circuit loop is zero.
In simple language, when you travel around a complete loop and return to your starting point, the total voltage rises and total voltage drops balance each other.
For a loop with one voltage source and several voltage drops:
Understand KVL using a battery and resistors
The battery raises the electric potential by 12 V. The two resistors cause voltage drops of 7 V and 5 V. Returning to the starting point gives:
KVL Solved Numerical: Find the Unknown Voltage
Question: A closed loop contains a 15 V battery and three voltage drops. Two of the drops are 4 V and 6 V. Find the third voltage drop, assuming the stated drops are in the direction of the loop current.
Step 1: Apply Kirchhoff's Voltage Law.
Step 2: Simplify.
Step 3: Calculate the unknown voltage.
Sign Convention for Kirchhoff's Laws
The signs in a circuit equation depend on your chosen current directions and the direction in which you trace a loop. The important rule is to remain consistent.
For KCL
- Choose currents entering the junction as positive.
- Choose currents leaving the junction as negative.
- Alternatively, reverse these choices, provided you apply them consistently.
For KVL
- Choose a direction to trace the closed loop.
- Moving from a battery's negative terminal to its positive terminal gives a voltage rise.
- Moving from positive to negative gives a voltage drop.
- For a resistor, moving in the assumed current direction gives a drop of IR; moving against it gives a rise of IR.
- Add the signed voltage changes and set their sum to zero.
Interactive Practice: Try Kirchhoff's Laws
Change the numerical values below to check current conservation at a junction and voltage conservation around a loop. These are simple calculators for practice.
Practice 1: KCL current calculator
Practice 2: KVL voltage calculator
Difference Between KCL and KVL
| Feature | KCL | KVL |
|---|---|---|
| Full name | Kirchhoff's Current Law | Kirchhoff's Voltage Law |
| Main focus | Current at a junction | Voltage around a closed loop |
| Basic principle | Conservation of electric charge | Conservation of energy |
| Formula | ΣI = 0 | ΣV = 0 |
| Useful for | Finding unknown branch currents | Finding unknown voltages and loop currents |
Applications of Kirchhoff's Laws
Kirchhoff's laws are useful for:
- Calculating unknown currents in multi-branch circuits.
- Finding voltage drops across electrical components.
- Analysing circuits with several voltage sources.
- Solving electrical network problems using nodal and mesh analysis.
- Checking circuit calculations and troubleshooting electronic systems.
Limitations and Conditions
Kirchhoff's laws are highly useful, but their ordinary circuit-analysis form relies on suitable assumptions. The circuit is generally treated as a lumped network, where voltages and currents can be assigned to individual components and connections.
- KCL: The sum of conduction currents at a node is equal when there is no net charge accumulation. In situations involving time-varying charge, displacement current and other electromagnetic effects may need to be included.
- KVL: The sum of voltage changes around a loop is zero in the ordinary lumped-circuit model. A changing magnetic flux linking the loop can induce an electromotive force, so the simple form must be extended to account for that effect.
- At very high frequencies or when circuit dimensions are comparable to signal wavelengths, distributed electromagnetic effects may make simple lumped circuit models insufficient.
Frequently Asked Questions
What are Kirchhoff's two laws?
Kirchhoff's Current Law (KCL) states that total current entering a junction equals total current leaving it. Kirchhoff's Voltage Law (KVL) states that the algebraic sum of voltage changes around a closed loop is zero.
What is the formula for Kirchhoff's Current Law?
The formula is ΣI = 0 when currents entering and leaving a junction are assigned opposite signs. Equivalently, total incoming current equals total outgoing current.
What is the formula for Kirchhoff's Voltage Law?
The formula is ΣV = 0. This means that all voltage rises and drops, with their signs included, add up to zero around a closed loop.
Why is Kirchhoff's Current Law based on conservation of charge?
Electric charge cannot accumulate indefinitely at an ordinary circuit junction in the steady-state model. Therefore, the amount of charge arriving per unit time equals the amount leaving per unit time.
Why is Kirchhoff's Voltage Law based on conservation of energy?
In a closed lumped circuit without an unaccounted time-varying magnetic flux, a charge completing a full loop returns to its starting potential. The energy gained and lost per unit charge balances.
Can Kirchhoff's laws be used in AC circuits?
Yes. Kirchhoff's laws can be applied to AC circuits using instantaneous quantities or phasors, provided the circuit model and electromagnetic assumptions are appropriate.
What is the difference between a junction and a loop?
A junction is a connection where branches meet. A loop is a closed path that returns to its starting point. KCL is applied at junctions, while KVL is applied around closed loops.
Test Your Understanding
Question 1: At a junction, 8 A enters. If 3 A and 2 A leave through two branches, how much current leaves through the third branch?
Question 2: A battery supplies 20 V to a loop. Two voltage drops are 8 V and 5 V. Find the third voltage drop.
Summary
Kirchhoff's laws provide two complementary rules for solving electrical circuits. KCL keeps track of current at junctions, while KVL keeps track of voltage changes around closed loops.
Practise identifying junctions, choosing loop directions, and applying the correct signs to make circuit calculations easier and more reliable.