“Know how to solve every problem that has been solved.” “What I cannot create, I do not understand.” — Richard Feynman

KCL, KVL, and Ohm's Law

Circuits

All of linear circuit analysis reduces to two laws and one algebraic identity. The laws describe how voltages and currents distribute through a network of components. The identity describes what happens at a single resistor. Once you've internalized them, any resistive circuit is a linear system of equations — write down current balance at every node and voltage balance around every loop, and the unknown voltages and currents fall out.

Charge, current, voltage

Current is the rate of charge flow through a wire, measured in amperes (1 A = 1 C/s). Voltage is the electric potential difference between two points, measured in volts (1 V = 1 J/C). The product of voltage times current is power (watts).

I = \frac{d Q}{d t}, V_{ab} = ϕ_{a} - ϕ_{b}, P = V I .

A circuit is a network of components connected by ideal wires. Components have two or more terminals; at each terminal there's a voltage (relative to a reference, usually ground) and a current (flowing in or out). The whole problem is to find every voltage and current in the network given the components and an applied source.

Ohm's law: the algebraic identity

A resistor enforces a linear relationship between the voltage across its terminals and the current through it:

V = I R,

where $R$ is the resistance in ohms (1 Ω = 1 V/A). Ohm's law isn't a fundamental law of physics — it's an empirical relationship that holds for ohmic materials (most metals, doped semiconductors at small bias) over the operating range of a real resistor. Diodes don't obey it. Inductors and capacitors don't obey it (they have their own laws, involving time derivatives). But for the pile of resistors that makes up most of a circuit board, this one equation describes the entire component.

Kirchhoff's current law (KCL)

At every node in a circuit, the sum of currents flowing in equals the sum of currents flowing out:

in \sum I_{k} = out \sum I_{k}, equivalently node \sum I_{k} = 0.

The physical content is charge conservation. A wire doesn't accumulate charge (it has no capacitance — that's a separate component), so anything that flows in must flow out, instantaneously. KCL is just charge-in = charge-out at every junction, written as a current balance.

Kirchhoff's voltage law (KVL)

Around any closed loop in a circuit, the sum of voltage drops is zero:

\oint_{loop} V_{k} = 0.

The physical content is that the electric potential is single-valued: if you walk around any closed loop and add up the voltage changes, you have to return to the same potential you started at. (For circuits in the quasi-static regime — small compared to the wavelength of any time-varying field — this is a consequence of $\nabla \times E = 0$ . Once you're at high frequencies and the inductive coupling between loops matters, KVL needs modification.)

Why these three are enough

For a circuit with $N$ nodes and $B$ branches (components), KCL gives $N - 1$ independent equations (one per node, minus one for the ground reference), KVL gives $B - N + 1$ independent equations (one per independent loop), and Ohm's law (plus the constitutive laws of any other components) gives $B$ more — one per component, relating its voltage and current. Total: $(N - 1) + (B - N + 1) + B = 2 B$ equations in $2 B$ unknowns (a voltage and a current per branch). The system is square, linear, and solvable.

That's the structural reason circuits are tractable. Every analog circuit, no matter how complicated, reduces to a square linear system in the resistive case — and to a square system of linear ODEs in the dynamic case (when capacitors and inductors enter). Modern circuit simulators (SPICE, Spectre, ngspice) use a variant called Modified Nodal Analysis that builds this system automatically and hands it to a sparse linear solver.

Two cases you'll see everywhere

Series. Two resistors in series share the same current. By KVL the total voltage is the sum of voltage drops:

R_{series} = R_{1} + R_{2} .

Parallel. Two resistors in parallel share the same voltage. By KCL the total current is the sum of branch currents:

\frac{1}{R _{parallel}} = \frac{1}{R _{1}} + \frac{1}{R _{2}} .

Voltage divider. Two resistors in series, source on one end, ground on the other, output tapped between them. By the series rule, the current is $I = V_{in} / (R_{1} + R_{2})$ , and the output voltage is the drop across $R_{2}$ :

V_{out} = V_{in} \cdot \frac{R _{2}}{R _{1} + R _{2}} .

Half the analog circuits in the world are voltage dividers somewhere. Reference generation, signal attenuation, biasing — it's all variations on this two-resistor pattern.

What's next

These laws cover every resistive circuit — every circuit made of sources and resistors, where the response to a voltage is instantaneous. The interesting circuits have memory: capacitors that store charge, inductors that store flux. Those components introduce time derivatives into the constitutive law, which turns the linear system into a linear system of ODEs. That's where most of the actual physics happens: time constants, transient responses, filters, oscillators. The RC circuit is the simplest example and the natural next page — it's also, as it turns out, the same equation that governs a leaky integrate-and-fire neuron.