Series and Parallel Circuits

Resistors connected end-to-end form a series circuit^[1]. Current has only one path through them, so the same current flows through every resistor and the voltages add up.

The total resistance of resistors in series is simply their sum:

$$R_{total} = R_1 + R_2 + R_3 + \cdots$$

Look at the voltage labels in the series circuit above: 12 V → 10 V → 6 V → 0 V. It's tempting to read the final 0 V as "all the power got used up." That's not quite right — and untangling it gives you the single most important rule in circuit analysis.

Principle — in any closed loop, the voltage drops across the components must sum to the source voltage. Every volt the battery supplies has to be claimed by something on the way back to the − terminal. Nothing is "left over." This is Kirchhoff's Voltage Law (KVL), and it's the rule that decides what every component in a circuit ends up doing.

A useful mental picture: the battery lifts charge up to a height of 12 V, and each resistor is a step the charge falls down. The total fall has to equal the total lift, so by the time the charge re-enters the battery, it's back at ground level — energy spent, ready to be lifted again.

Voltage is always a difference between two points, never a quantity sitting at one point. The labels in our series diagram are the potential at each node measured against the − terminal, and the − terminal is the reference we chose to call zero. The "0 V" at the end of the chain just means "this point is on the same wire as the − terminal," which it literally is.

The chart below makes this concrete. Each colored block is a resistor, and its height = the voltage drop across it. Slide the resistors and watch the blocks resize: the drops always sum to 12 V, and each resistor claims a share proportional to its resistance. A resistor that's twice as big drops twice as much voltage — because the same current flows through every component in a series loop, so V = I·R makes the bigger R take the bigger share.

Now apply that same KVL picture to a circuit you've probably actually built: an LED with a current-limiting resistor.

An LED is a stubborn component. Once it's lit, it sits at a roughly fixed forward voltage — about 2 V for a red one, 3.2 V for blue. So if you wire a 2 V red LED straight across a 5 V supply, you have a contradiction: the LED says "I'll be at 2 V," the supply says "I'm 5 V," and somebody has to claim the leftover 3 V. With nothing else in the loop, that 3 V falls across the LED's and supply's tiny internal resistance, and I = V/R with a tiny R means a huge current. The LED cooks.

Add a resistor in series and the leftover 3 V has somewhere reasonable to go. KVL says:

$$V_{source} = V_{resistor} + V_{LED}$$

so the resistor takes whatever the LED doesn't, and the current that flows through both is set by Ohm's law on the resistor:

$$I = \frac{V_{source} - V_f}{R}$$

The resistor isn't really "limiting voltage on the LED" — it's there to soak up the leftover voltage in a way that produces a safe current. Hence "current-limiting resistor."

Try it below. Pick a supply voltage, pick a resistor, pick an LED color (each color has its own forward voltage). Watch the voltage staircase split between the resistor and the LED, and watch the current flag the LED as off / dim / nominal / fried.

When resistors are connected across the same two nodes — each one bridging between them — they form a parallel circuit. The voltage across each resistor is the same, and the currents add up.

For parallel resistance, take the sum of reciprocals:

$$\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \cdots$$

The equivalent resistance is always smaller than the smallest individual resistor.

Knowing how resistance combines is what lets you reason about a real circuit instead of staring at a wall of components:

• Christmas lights and old LED strings. Bulbs wired in series all share the same current — when one burns open, the entire string goes dark. Modern strings use parallel branches or shorting filaments to avoid this failure mode.
• Household wiring. Every outlet on a circuit is in parallel, so each gets the full mains voltage independently. The circuit breaker watches the total current — the sum of every device drawing in parallel — and trips when that exceeds the breaker's rating.
• Speaker loads. Two 8 Ω speakers in parallel present a 4 Ω load to the amplifier, doubling the current the amp must source. Many amps overheat or shut down. Wire them in series instead and you get 16 Ω — safer for the amp, but quieter.
• Solar panels. Cells are wired in series within each panel to add their voltages, then panels run in parallel to add their currents. A single shaded cell in a series string drops the entire string's output, which is why panel layouts matter.

Series-vs-parallel reasoning is the bridge between "I know Ohm's Law" and "I can read a real schematic."