Series & Parallel Circuits

Two ways to wire, two very different results

Junior level — plain language, no maths

Take a battery and two identical light bulbs. There are two ways to connect them, and they behave in surprisingly opposite ways. Wire the bulbs one after the other, in a single loop, and you have a series circuit. Wire each bulb on its own separate branch, side by side, and you have a parallel circuit. Same battery, same bulbs — completely different results.

In series, the same current must thread through both bulbs, one then the other, like beads on a single string. Because the two bulbs share the push of one battery between them, each gets only half, so both glow dimly. Worse, if one bulb blows, the loop is broken and both go dark — which is exactly how old Christmas lights infuriated everyone: one dud killed the whole string.

In parallel, each bulb gets its own path straight back to the battery, so each one feels the battery's full push and shines brightly. Unscrew one and the other keeps glowing, because its path is untouched. This is why your home is wired in parallel: every lamp and appliance gets the full voltage, and switching off the kettle doesn't plunge the whole house into darkness. In the simulation below, flip between the two and watch the bulbs.

Adding resistances, and how current and voltage split

Student level — the core equations

The two wirings obey mirror-image rules. In series, there is only one path, so the same current \(I\) flows everywhere, and the resistances simply add: \(R_{\text{series}} = R_1 + R_2\). The battery's voltage divides between the components in proportion to their resistance. Two \(10\ \Omega\) bulbs on a \(6\ \text{V}\) battery give \(20\ \Omega\) total and a current of just \(0.3\ \text{A}\), with \(3\ \text{V}\) dropped across each bulb.

In parallel, every branch feels the full battery voltage, and the currents add instead. The resistances combine reciprocally: \(\dfrac{1}{R_{\text{parallel}}} = \dfrac{1}{R_1} + \dfrac{1}{R_2}\), which always gives a total smaller than either branch. Those same two \(10\ \Omega\) bulbs now make just \(5\ \Omega\), pulling \(1.2\ \text{A}\) from the battery — four times the series current — with each bulb getting the full \(6\ \text{V}\) and glowing far brighter.

Brightness follows power, \(P = VI = I^2R\). In series each bulb dissipates \(I^2R = 0.3^2\times10 \approx 0.9\ \text{W}\); in parallel each burns \(0.6^2\times10 = 3.6\ \text{W}\), four times as much. That factor of four is exactly what you see as the jump in brightness. It also warns why parallel branches can overload a supply: add more of them and the total current keeps climbing, which is what trips a household breaker.

Kirchhoff's laws, real sources and network reduction

Scholar level — full mathematical depth

01The two conservation laws behind every circuit

Series and parallel are just special cases of two deeper rules. Kirchhoff's current law says charge is conserved at any junction — currents in equal currents out, \(\sum I = 0\) — which is why parallel branch currents add. Kirchhoff's voltage law says energy is conserved around any loop — the voltage rises and drops sum to zero, \(\sum V = 0\) — which is why series voltages divide. Every circuit analysis, however tangled, reduces to applying these two statements.

02Collapsing a network

With those laws you can shrink any resistor network by repeatedly merging series and parallel groups into equivalents, until a single resistance remains. Circuits that resist this — the classic Wheatstone bridge — need the full simultaneous equations (node or mesh analysis), which is really just Kirchhoff's laws written as linear algebra. The bridge is not a curiosity: balanced, it measures an unknown resistance with exquisite precision, and it underlies strain gauges and countless sensors.

03No battery is ideal

Real sources have internal resistance \(r\), so the voltage you actually get sags under load: \(V_{\text{terminal}} = \varepsilon - Ir\). This is why a car's headlights dim for an instant when the starter motor — a huge, low-resistance load — draws hundreds of amps, and why a nearly-flat battery (whose \(r\) has climbed) can read a healthy voltage yet collapse the moment you ask it for current. The ideal battery of the diagrams is always an approximation.

04Maximum power, and its cost

How much load extracts the most power from a real source? The maximum power transfer theorem says power delivered peaks when the load resistance equals the source's internal resistance, \(R_L = r\). But at that match only half the energy reaches the load; the rest is wasted heating the source. Power electronics therefore deliberately avoids the matched condition — the grid runs \(R_L \gg r\) for efficiency — while radio and audio engineers embrace matching, where transferring the signal matters more than wasting a little power.

05When components remember: transients

Add a capacitor or inductor and the circuit gains memory of time. An RC circuit charges and discharges on a timescale \(\tau = RC\), following \(V(t) = V_0(1 - e^{-t/\tau})\); an RL circuit does the analogous thing with \(\tau = L/R\). These exponential transients set how fast a circuit can switch, filter which frequencies pass, and — chained together — become the timing elements and filters at the heart of every electronic device.

06Beyond lumped elements

All of this assumes lumped components joined by resistanceless wires — an approximation that holds while the circuit is far smaller than the signal's wavelength. Push to high frequencies and the wires themselves become transmission lines with their own impedance, signals reflect off mismatches, and Kirchhoff's tidy laws give way to Maxwell's equations. The humble series-parallel distinction is the first rung of a ladder that climbs all the way to microwave engineering and the design of every fast digital chip.