Learn Kirchhoff’s current law (KCL) and voltage law (KVL) with simple explanations, circuit diagrams, sign rules, a worked example and a detailed FAQ.
Kirchhoff’s Laws (KCL and KVL) Explained with Circuit Diagrams and Examples
Kirchhoff’s laws are the two basic rules used to analyse almost every electric circuit. Kirchhoff’s Current Law (KCL) says that the current entering a junction equals the current leaving it. Kirchhoff’s Voltage Law (KVL) says that the voltages around any closed loop add up to zero. This guide explains both laws in plain language, with diagrams, sign rules, a worked example, common mistakes and a detailed FAQ.
Who Was Gustav Kirchhoff?
Gustav Robert Kirchhoff was a German physicist who published these two circuit laws in 1845, when he was a student. They extend Ohm’s law to circuits with several branches, sources and loops. They are still the starting point of circuit analysis in electronics and electrical engineering.
Basic Terms You Need First
- Node (junction): a point where two or more circuit elements connect.
- Branch: a single element, or a series path, between two nodes.
- Loop: any closed path that starts and ends at the same node without visiting a node twice.
- Mesh: a loop that contains no other loop inside it.
Kirchhoff’s Current Law (KCL)
Statement
The algebraic sum of all currents meeting at a node is zero. Equivalently, the total current flowing into a node equals the total current flowing out of it.
Formula: ΣI(in) = ΣI(out), or ΣI = 0
Why it is true
KCL comes from the conservation of electric charge. Charge cannot pile up or disappear at a junction in a normal circuit, so whatever flows in must flow out.
Diagram 1: KCL at a node
I2 = 2 A (out)
↗
I1 = 5 A (in) ●
───────────────→ (node)
↘
I3 = 3 A (out)
KCL: I1 = I2 + I3 → 5 A = 2 A + 3 A
Sign convention
Choose one rule and keep to it. Currents entering the node are positive and currents leaving are negative, or the reverse. For Diagram 1: +5 − 2 − 3 = 0.
Kirchhoff’s Voltage Law (KVL)
Statement
The algebraic sum of all voltages around any closed loop is zero. Put another way, the voltage rises from sources equal the voltage drops across the other components.
Formula: ΣV = 0, or ΣV(rise) = ΣV(drop)
Why it is true
KVL comes from the conservation of energy. A charge that travels all the way round a loop returns to its starting point. So the total energy it gains from sources must equal the energy it loses in the components.
Diagram 2: KVL in a series loop
R1 = 2 Ω (drop 2 V)
┌────[█████]────→────┐
│ I = 1 A │
│ │
+ │ [██] R2 = 4 Ω
12 V│ [██] (drop 4 V)
− │ │
│ │
└────[█████]─────────┘
R3 = 6 Ω (drop 6 V)
KVL: 12 − 2 − 4 − 6 = 0
Sign convention
Pick a direction round the loop; clockwise is common. Going through a source from − to + is a rise (+). Going through a resistor in the direction of the assumed current is a drop (−). Reverse these signs if you go against the current.
KCL vs KVL at a Glance
| Feature | KCL (Current Law) | KVL (Voltage Law) |
|---|---|---|
| Applies to | Nodes (junctions) | Closed loops or meshes |
| Equation | ΣI = 0 | ΣV = 0 |
| Based on | Conservation of charge | Conservation of energy |
| Main method | Nodal analysis | Mesh (loop) analysis |
| Solves for | Branch currents, node voltages | Loop currents, voltage drops |
How to Apply Kirchhoff’s Laws: Step by Step
Diagram 3: Analysis flow
┌────────────────────────────────────────┐
│ 1. Label nodes, loops and currents │
└───────────────────┬────────────────────┘
▼
┌────────────────────────────────────────┐
│ 2. Mark polarity (+/−) on each element │
└───────────────────┬────────────────────┘
▼
┌────────────────────────────────────────┐
│ 3. Write KCL and KVL equations │
└───────────────────┬────────────────────┘
▼
┌────────────────────────────────────────┐
│ 4. Solve the equations together │
└───────────────────┬────────────────────┘
▼
┌────────────────────────────────────────┐
│ 5. Check: KCL and KVL must hold │
└────────────────────────────────────────┘
- Label the circuit. Name every node and give each branch a current with an assumed direction. A wrong guess is fine, because it will simply give a negative answer.
- Mark polarities. Put + and − on each resistor so that current enters at +.
- Write equations. Apply KCL at independent nodes and KVL around independent loops. You need as many equations as unknowns.
- Solve. Use substitution, elimination or matrices.
- Check. Confirm that the currents balance at every node and the voltages balance around every loop.
Worked Example (KCL and KVL Together)
Two sources feed one node A. Source V1 = 12 V connects through R1 = 2 Ω. Source V2 = 6 V connects through R2 = 3 Ω. R3 = 6 Ω connects from A to ground. Find the three currents.
V1 = 12 V ──[R1 = 2 Ω]──┐
├── A ──[R3 = 6 Ω]── ground
V2 = 6 V ──[R2 = 3 Ω]──┘
Step 1: KCL at node A. I1 and I2 flow toward A, and I3 flows to ground.
I1 + I2 = I3
Step 2: Write each current using the node voltage VA.
(12 − VA)/2 + (6 − VA)/3 = VA/6
Step 3: Solve. Multiply by 6: 3(12 − VA) + 2(6 − VA) = VA. This gives 48 = 6VA, so VA = 8 V.
Results
- I1 = (12 − 8)/2 = 2 A
- I2 = (6 − 8)/3 = −0.67 A
- I3 = 8/6 = 1.33 A
The negative sign means I2 actually flows into source V2, so that source is being charged.
Check: KCL gives 2 − 0.67 = 1.33 A, which matches I3. KVL on the loop through V1, R1 and R3 gives 12 − (2 × 2) − (1.33 × 6) = 12 − 4 − 8 = 0.
Applications of Kirchhoff’s Laws
- Circuit analysis methods such as nodal analysis, mesh analysis and superposition.
- Designing voltage dividers, current dividers and bridge circuits such as the Wheatstone bridge.
- Power distribution, battery packs and electronic board design.
- Circuit simulators such as SPICE, which solve KCL and KVL equations for every node and loop.
- Fault finding: a measured current or voltage that breaks KCL or KVL points to a wiring error or a faulty part.
Common Mistakes to Avoid
- Mixing sign conventions halfway through a problem.
- Forgetting that a resistor’s polarity depends on the current direction you assumed.
- Writing dependent equations, such as KCL at every node of a circuit. One node equation is always redundant.
- Treating a negative answer as an error instead of a reversed current direction.
Limitations of Kirchhoff’s Laws
Kirchhoff’s laws apply to lumped circuits, where component sizes are much smaller than the signal wavelength. At very high frequencies, in long transmission lines, or where a changing magnetic field passes through a loop, KVL needs extra terms or a different method, such as Maxwell’s equations or distributed models.
Frequently Asked Questions About Kirchhoff’s Laws
What are Kirchhoff’s laws?
They are two rules for circuit analysis. Kirchhoff’s Current Law (KCL) states that the currents at a node sum to zero. Kirchhoff’s Voltage Law (KVL) states that the voltages around a closed loop sum to zero.
What is KCL in simple words?
Whatever current flows into a junction must flow out of it. Nothing is stored or lost at the junction, so 5 A in means 5 A out, split between the outgoing wires.
What is KVL in simple words?
If you walk once around any loop in a circuit, the voltage you gain from batteries equals the voltage you lose across resistors and other parts. The total change is zero.
Which physical principles are Kirchhoff’s laws based on?
KCL is based on conservation of electric charge. KVL is based on conservation of energy, in the lumped-circuit approximation.
How do I choose signs in KVL?
Choose a direction round the loop. A rise (moving from − to + through a source) is positive. A drop (moving along the current through a resistor) is negative. Keep the same rule for the whole loop.
What does a negative current mean in my answer?
It means the real current flows opposite to the direction you assumed. The calculation is still correct, and you can keep the negative sign in later steps.
Do Kirchhoff’s laws work for AC circuits?
Yes, at normal power and audio frequencies. For AC you apply them to instantaneous values or to phasors, using impedance instead of resistance.
What is the difference between Ohm’s law and Kirchhoff’s laws?
Ohm’s law (V = IR) relates voltage and current in a single resistor. Kirchhoff’s laws describe how currents and voltages combine across a whole network, so you usually need both to solve a circuit.