How to engineer.
Electronics explained with water — from a barrel in the sky to a transistor on your bench. The whole course, on paper.
This book is the companion to the free Fathohm course at fathohm.com. Everything here is also on the screen, with live animations and sliders. Paper is for the desk, the pencil, and the margin note.
Read it in order or wander. Each chapter stands alone, but they stack: pressure before flow, flow before resistance, all three before everything else.
Two voices run through it. The plain path is the main text — read that and you'll understand. The formula boxes and worked examples are there when you want the numbers; skip them on the first pass if you like. Nothing important hides in an equation.
You will not be asked to memorize. You'll be asked to picture — water in a pipe, a bucket filling, a wheel spinning down — and trust that the invisible electricity does the same thing.
Not literally. But the math is the same, the intuition is the same, and once you see the resemblance you'll never un-see it.
A wire is a pipe. A battery is a barrel of water held up high. A resistor is a pinch in the pipe. The thing we actually care about — the thing that does work at the far end — is the flow.
Anything water does, electricity does too. So throughout this book we'll look at the water and trust that the wire is doing the same thing, silently, at nearly the speed of light.
| Water pressure | → Voltage V |
| Rate of flow | → Current I |
| A pinch in the pipe | → Resistance R |
| Opening the tap | → Closing the circuit |
Ten ideas, each one a thing water does. Learn these and you can read almost any circuit.
Three quantities run all of electronics, and they're the three things you already understand about water.
Voltage is the push. Picture a barrel of water held up high. The taller the column, the harder it presses down on the pipe. A AA battery is a 1.5 V barrel; a car battery, a 12 V reservoir; a wall outlet, a 120 V tower. Voltage is potential — the push that could happen, whether or not anything is flowing.
Current is the flow. Open the tap and water moves; that rate of movement is current, measured in amps. In a wire it's the count of electrons passing a point each second — about 6.24 × 10¹⁸ of them per amp, per second. You don't set current with a knob. It falls out of how hard you push against how much you pinch.
Resistance is the pinch. Squeeze the pipe and less gets through, no matter how hard you push. Every wire has a little; every component has its own. A toaster's wire has a lot — and that's why it gets hot.
A 9 V battery pushes through a 450 Ω resistor. How much flows?
I = V / R = 9 / 450 = 0.02 A = 20 mA — a typical, healthy current for a small LED.
When the pipe splits or runs in a line, two simple rules tell you everything.
In series, parts sit one after another on a single path — like two pinches in the same pipe. The same flow goes through every one, and their resistances add up. More parts in a row, more total pinch, less current overall.
In parallel, the pipe splits into branches that rejoin. Each branch gets the full push, so adding branches gives the water more ways through — total resistance goes down, and total flow goes up. Two identical branches each carry half the current; the source sees double.
Two 100 Ω resistors. In series: 200 Ω. In parallel: 50 Ω.
Same two parts — wiring decides whether you double the pinch or halve it.
Push times flow is power. Power, in a resistor, comes out as heat — and that's not a bug.
Voltage is the push and current is the flow; multiply them and you get power — the rate energy is delivered, measured in watts. A waterwheel turns faster when there's both height and volume behind the water. Lots of push but a trickle of flow does little work; so does a flood with no pressure. You need both.
In a resistor, that power becomes heat. It's why a toaster glows, why a phone charger is warm, and why every part has a limit. Pick a resistor too small for the job and you'll find out — with a wisp of smoke.
That 450 Ω resistor at 20 mA: P = I²·R = (0.02)² × 450 = 0.18 W.
Under ¼ watt — a standard resistor handles it with room to spare.
A capacitor is a bucket in the pipe. It fills with charge, holds it, and gives it back.
Put a small bucket across the flow and it fills up over time — fast at first, then slower as it nears the top. While it's filling, current flows into it; once full, the flow stops. Disconnect the push and the bucket pours its charge back out. That's a capacitor: it stores energy in an electric field and releases it on demand.
How long it takes to fill depends on the bucket's size and the pinch feeding it. Big bucket or tight pinch — slow fill. This fill-time is the secret behind every delay, every smooth power supply, and every blink.
10 kΩ feeding a 100 µF bucket: τ = 10 000 × 0.0001 = 1 s.
So it's ~63% charged after 1 second, and full after about 5.
The simplest part is the most profound. A switch is a tap — and a tap that a machine can flip is a thought.
A switch opens or closes the pipe. Closed, water flows and the loop is complete; open, everything stops. There's no in-between that matters — it's all or nothing. That two-state, yes-or-no nature is exactly what we call binary: on or off, 1 or 0.
Stack switches in a row and the water only flows if all are closed — that's an AND. Put them side by side and flow gets through if any is closed — that's an OR. Those two patterns, repeated a billion times on a chip, are how a computer reasons.
The most important invention of the century is a tap that water turns on by itself.
Imagine a tap where a trickle of water in a side pipe pushes open a flood in the main pipe. A small flow controls a large one. That's a transistor: a little current at its control leg (the base) lets a much bigger current run through the other two (collector to emitter).
Two superpowers fall out of that. As an amplifier, a faint signal at the base becomes a strong copy in the main flow. As a switch, a control signal turns the main current fully on or off — no human finger required. Because one circuit can now flip another, machines can think.
A diode is a one-way valve in the pipe. Push forward and water flows; push back and it slams shut.
A diode lets current pass in one direction and blocks it in the other — exactly like a check valve. It costs a small, fixed push to open (about 0.7 V for a silicon diode, a little less for an LED's red glow), and above that it conducts freely. Reverse the push and nothing gets through at all.
That one-way behavior is how you protect a circuit from a backwards battery, how you steer current where it should go, and — pointed cleverly — how you turn the back-and-forth of AC into the steady push of DC. An LED is just a diode that spends its forward push as light.
Some sources push one steady direction. Others slosh the water back and forth many times a second.
DC — direct current — is water pushed steadily one way, like a barrel draining downhill. A battery gives DC. It's simple, calm, and what most of your circuits run on.
AC — alternating current — sloshes back and forth instead, reversing many times each second. The wall outlet is AC, swinging 50–60 times a second (50–60 Hz). Why bother? Because AC's pressure can be stepped up enormously high for efficient travel across the country, then stepped back down — something DC can't do as easily. That's why the grid is AC and your gadgets quietly convert it back to DC.
An inductor is a heavy paddlewheel in the pipe. It hates to start, and once spinning, hates to stop.
Drop a weighty waterwheel into the flow. When you first push, it resists — all your effort goes into getting it turning, and current rises only slowly. But once it's spinning, it carries momentum: cut the push and the wheel keeps the water moving for a moment, even forcing it along.
An inductor does exactly this with current. It opposes changes in flow — slow to rise, slow to fall. That stored momentum lets it smooth bumpy currents, and, paired with a capacitor, ring back and forth like a pendulum. Where a capacitor resists changes in voltage, an inductor resists changes in current — perfect mirror images.
A bucket that fills and a switch that flips at the brim. Repeat forever, and you have a clock.
Here's the trick that runs every blinker, beeper, and clock. Let a capacitor fill through a resistor (Chapter 4's bucket and pinch). Watch it with a switch that flips when the bucket hits a line, dumps it, and lets it refill. Up, down, up, down — a steady rhythm born from parts that, alone, just sit there.
Build it from scratch with two transistors and you get the classic flip-flopping blinker. Or reach for the 555 timer — a chip that packages the whole idea: three resistors set the ⅓ and ⅔ marks, two comparators watch the bucket, and a flip-flop runs the output. The cap sawtooths between ⅓ and ⅔ of the supply, forever.
Theory you can hold. A bench, a handful of parts, and one device that grows chapter by chapter.
Your workbench is a plastic block full of holes. Three rules and you can build anything in this book.
A column is one wire. The five holes stacked in each little column are joined together inside the board. Drop two legs into the same column and they're connected — no solder, no fuss.
The side rails carry power. The long + and − stripes down each edge run the full length of the board. Feed them from your battery once, and every column can tap in.
In one loop, order is free. The same current flows through everything in a series loop, so a resistor before or after the LED limits it identically. Position is tidiness, not function — a freeing thing to learn early.
The battery, LED, and diode have a + and − end — put them in the right way round. A resistor or jumper doesn't care which way it goes. When something won't light, check polarity first.
One device, built across the whole course — a flashing signal light you engineer part by part.
Every chapter quietly adds to one machine. You light an LED (Chapter 1), size its resistor so it doesn't burn out (Chapter 3), add more lights in series and parallel (Chapter 2), store charge to make it fade (Chapter 4), drive it harder with a transistor (Chapter 6), protect it with a diode (Chapter 7), and finally give it a heartbeat with a 555 (Chapter 10). The result is the Beacon: a bright, blinking signal that runs all night on a 9-volt battery.
A breadboard, three red LEDs, a 555 timer, an NPN transistor, a signal diode, an assortment of resistors and capacitors, two pushbuttons, a 9 V battery and clip, jumper wires, and a light sensor. A few dollars, all common parts — Adafruit, SparkFun, and DigiKey each stock everything.
Every term in one place, each tied back to the water.
Now go build something.
The whole course — animations, sliders, quizzes, and the Beacon build — is free at fathohm.com. This guide is yours to keep, mark up, and pass on.