Kirchhoff’s Laws (KCL & KVL) with Worked Examples

Ohm’s law handles one component. The moment a circuit has more than two nodes you need Kirchhoff. One law about current at junctions, one about voltage around loops. Together they solve every linear network.
> At a glance: 9 minute guide · part 4 of 10 in the electronics fundamentals complete guide track · includes a worked example and a quick-reference table.

## KCL: current in equals current out

At any node, charge cannot pile up. The sum of incoming currents equals the sum of outgoing currents. This is bookkeeping for electrons and it is why the current entering a series string equals the current leaving it. Why parallel branch currents add back to the total.

| L | a | w | | | | | | | | | | | |
| — | — | — | — | — | — | — | — | — | — | — | — | — | — |
| S | t | a | t | e | m | e | n | t | | | | | |
| C | o | n | s | e | r | v | e | s | | | | | |
| G | u | a | r | d | s | | a | g | a | i | n | s | t |
| KCL | Σ current in = Σ current out at a node | Charge | Impossible current “loss” at junctions | | | | | | | | | | |
| KVL | Σ voltages around any loop = 0 | Energy | Voltage drops that do not add up | | | | | | | | | | |
| Ohm | V = I × R per element | | Element-level inconsistency | | | | | | | | | | |

## KVL: voltage around a loop sums to zero

Tracing any closed loop, the sum of source voltages equals the sum of drops across the elements. Energy returned per charge must balance; batteries lift charge up, loads drop it back down. KVL is the reason voltage dividers and series strings behave so predictably.

## Solving a two-loop circuit step by step

Label branch currents, write KCL at the independent nodes, write KVL around each mesh, then solve the simultaneous equations. For two loops this is two equations in two unknowns five minutes by hand, instant by matrix.

## How to apply this in your build

Work through the sequence below each step assumes the previous one passed. For numbers that need calculating, the linked tools at the end of this guide do the arithmetic instantly.
1. Assign a current direction to every branch (guess if unsure)
2. Apply KCL at each node except ground
3. Walk each mesh writing KVL with sign conventions
4. Solve the equations; a negative answer simply flips your guessed direction

### Worked example

Node A feeds 1 A in; R1 leaves with 0. 4 A. By KCL, R2 must carry 0. 6 A. If both drop toward a shared 5 V rail through 10 Ω, drops are 4 V and 6 V KVL confirms 4 + 6 = 10 V supply. Run the numbers yourself with the series-parallel and the result should agree to within rounding.

> Practical note from the bench. Interview reality check: engineers who can KVL a two-loop divider on paper in under a minute are the ones who debug boards fastest.

## Common mistakes to avoid

– Dropping the sign of a voltage when walking the loop against current flow
– Forgetting to include source internal resistance in the loop
– Writing more equations than independent nodes redundant rows confuse solving

## Key takeaways

– KCL: current in equals current out the foundation of this guide. Revisit it if any measurement here surprises you.
– KVL: voltage around a loop sums to zero the foundation of this guide. Revisit it if any measurement here surprises you.
– Solving a two-loop circuit step by step the foundation of this guide. Revisit it if any measurement here surprises you.

## Prerequisites and preparation

Before starting. Assign a current direction to every branch (guess if unsure) and apply kcl at each node except ground. Keep the [series-parallel](/tools/series-parallel) open every number in the worked example is reproducible. Total time including the bench steps: about 7–9 minutes.

## Who benefits most

Hobbyists meeting this topic for the first time, students who want the version with real numbers instead of abstract symbols. Returning engineers refreshing a corner of the craft. The mistake list alone justifies the visit every entry in it was learned the expensive way.

### Quick reference card

| Aspect | Where to find it in this guide |
| — | — |
| Core theory | KCL: current in equals current out |
| Application steps | How to apply this in your build |
| Worked numbers | Worked example |
| Failure modes | Common mistakes to avoid |

## How this fits the electronics fundamentals complete guide track

This guide is one stop in the structured learning path. Start from the [electronics fundamentals complete guide](/tutorial/electronics-fundamentals-complete-guide) pillar page for the full map, or continue with [series and parallel rules](/tutorial/series-vs-parallel-circuits) and [resistivity of conductor materials](/tutorial/resistance-and-resistivity-guide). For the arithmetic, open the [series-parallel](/tools/series-parallel).

## Frequently asked questions

Do Kirchhoff’s laws work in AC circuits?
Yes, with phasor sums. The same node and loop bookkeeping applies; reactances replace resistances.

When should I use mesh vs nodal analysis?
Fewer loops → mesh; fewer nodes → nodal. Both give identical answers.

Is there a calculator for this?
Yes the [series-parallel](/tools/series-parallel) tool runs the formulas from this guide instantly, client-side, with no signup.

## Continue the learning path

– The complete electronics fundamentals guide: [Electronics Fundamentals complete guide](/tutorial/electronics-fundamentals-complete-guide)
– Read next: [diodes and transistors explained: the two semiconductor families](/tutorial/diodes-and-transistors-explained)
– Also in this track: [passive components: resistors, capacitors and inductors compared](/tutorial/passive-components)
– Continue with: [the pn junction diode: physics, curves and applications](/tutorial/pn-junction-diode)
– Calculate as you go: [Ohm’s law calculator](/tools/ohms-law) · [resistor colour code decoder](/tools/resistor-color-code) · [RC time constant tool](/tools/rc-time-constant)
– Bookmark this page against the day a measurement surprises you. Most readers return to the table and the mistake list first, and that is the correct order.

## Measurement discipline

Keep a lab notebook entry for every build in this track. The measured values, the deviations from the guide and the reason for each. Six months from now, those notes are worth more than any tutorial. They describe your bench and your components rather than a general case.

When a result here disagrees with your expectation, write down both numbers before changing anything. The gap between predicted and measured is where the real engineering lives. It is usually a tolerance, a parasitic or an assumption that was never checked.

## Hard-won notes

Component tolerance and meter accuracy stack. A 5% resistor, a 2% reference and lead resistance easily explain small gaps. Compare direction and magnitude before suspecting the guide.

A resistor and capacitor kit, common diodes and transistors, a breadboard and jumpers. Add modules as tracks demand them.