Kirchhoff's Two Laws
Start here. Break each law on purpose and watch charge pile up at a node and energy appear from nowhere round a loop.
Skip to the animationKirchhoff's laws say that the currents at any node sum to zero and the voltages round any closed loop sum to zero — which are conservation of charge and conservation of energy respectively, and between them they are the entire basis of circuit analysis.
The vocabulary, briefly
- Node
- A point where two or more elements connect. Everything joined by unbroken wire is the *same* node, however far apart it is drawn — a fact that removes half the difficulty from nodal analysis.
- Branch
- A single element between two nodes.
- Loop
- Any closed path through the circuit that returns to its starting point without passing through any node twice.
- Mesh
- A loop with nothing inside it. Every mesh is a loop; not every loop is a mesh.
KCL, and why it cannot be otherwise
Kirchhoff's Current Law: the algebraic sum of currents entering a node is zero. Equivalently, everything in equals everything out.
The reason is worth doing numerically. Suppose 8 A entered a node and only 6 A left. The missing 2 A would accumulate at a point — 2 coulombs after one second. A wire junction has a capacitance of femtofarads, so V = Q/C puts that at the order of 10¹¹ volts. In reality the imbalance is corrected within picoseconds by exactly the field that voltage implies.
So KCL is not a modelling assumption. It is conservation of charge applied to a point that can neither store nor create it, and a circuit that violated it would not be an unusual circuit but an impossible one.
In practice you assign each branch an assumed direction and write ΣI = 0. The assumed directions do not have to be correct. A current that actually flows the other way simply comes out negative — which is what makes it possible to analyse a circuit you have not solved yet.
KVL, and why it cannot be otherwise either
Kirchhoff's Voltage Law: the algebraic sum of potential differences round any closed loop is zero.
Potential is a property of a point. Walk a closed loop and you return to the point you started at, so you must return to the potential you started at. If the rises and drops did not cancel, that single point would hold two potentials at once — and carrying one coulomb round the loop would produce energy from nothing.
- 1Choose a starting point and a direction to walk. Either direction works.
- 2At each element, write down the sign of the terminal you reach first.
- 3Through a source
−to+is a rise: write+V. Through a resistor+to−is a drop: write−IR. - 4Set the total to zero and solve.
Walk the loop the other way and every single term changes sign. The equation is unchanged, because −0 = 0. This is the reassurance worth having: the direction cannot be wrong, only inconsistent.
What is built directly on top
| Method | What it actually is |
|---|---|
Series resistance R = R₁ + R₂ | KVL round a loop with one current |
Parallel resistance 1/R = 1/R₁ + 1/R₂ | KCL at a node with one voltage |
| Voltage divider | KVL, solved for one element's share |
| Current divider | KCL, solved for one branch's share |
| Nodal analysis | KCL written at every node but the reference |
| Mesh analysis | KVL written round every mesh |
| Thevenin, Norton, superposition | The two laws plus linearity — consequences, not new physics |
This is why a result that appears to contradict Kirchhoff has always been misapplied rather than discovered. The laws are upstream of every technique in the syllabus.
Where the laws stop being exact
Both laws are the lumped element approximation of Maxwell's equations, and that approximation has a stated condition: the circuit must be small compared with the wavelength of the signals in it.
- KVL assumes no changing magnetic flux passes through the loop. A transformer deliberately breaks that, which is why its coupling is modelled as an explicit element rather than left implicit.
- KCL assumes no charge accumulates. At high frequency, stray capacitance to ground is a real branch, and ignoring it is why a circuit works at 1 kHz and misbehaves at 1 GHz.
- At microwave frequencies the approximation fails entirely, and transmission line theory replaces it — the same shift the power systems subject makes for long lines.
The numbers you will be asked for
- Kirchhoff's Current Law
Σ I = 0 at every node
Conservation of charge. Sign convention: in positive, out negative.
- Kirchhoff's Voltage Law
Σ V = 0 round every closed loop
Conservation of energy. Sign convention: the terminal you reach first.
- Ohm's law
V = I·R
An element law, not a Kirchhoff law — it describes one component, not the network.
- Independent equations
KCL: n − 1 equations · KVL: b − n + 1 equations
For n nodes and b branches — exactly enough to solve for every branch current.
Advantages and disadvantages
Advantages
- Two laws cover every lumped circuit, whatever the elements are.
- They hold for AC, DC, transient and non-linear circuits alike.
- Assumed current directions need not be correct — the algebra reports the truth as a sign.
- They reduce circuit analysis to writing and solving simultaneous equations.
Disadvantages
- Both assume lumped elements, and fail when the circuit is comparable in size to a wavelength.
- KVL is invalid for a loop with changing flux through it unless the coupling is modelled explicitly.
- Large networks produce large systems of equations, which is why the theorems in the next module exist.
- The laws say nothing about what the elements themselves do; every element still needs its own equation.
Watch it work
Check yourself
question 1 / 5
One question at a time. Pick an answer to see why it is right or wrong, then move on — there is no score to keep and nothing is saved.