A toroid is a ring of iron wrapped in copper — and it has one stubborn rule: it ignores anything that holds steady and passes only what moves. You cannot send it a level. You can only send it edges. Which is exactly why the language for talking through one is pulse-width modulation — information written into the timing of edges.
If you read the capacitor series, this will feel like looking in a mirror. There, a capacitor turned a changing voltage into a current: i = C·dv/dt. A toroid is the magnetic dual — it turns a changing flux into a voltage: v = N·dΦ/dt, Faraday's law, where N is the number of turns of wire and Φ is the magnetic flux threading the ring.
Drive a steady current through the winding and a steady flux sits in the core — unchanging, and so it induces nothing: a constant flux has zero rate of change, so v = 0. The toroid is deaf to a held level. But move that current — switch it, pulse it, edge it — and the flux moves with it, and a voltage appears in exact proportion to how fast it changed. Faster edge, bigger voltage.
A toroid cannot feel a current that holds still. It only ever answers the moment the current changed.
That single fact decides everything downstream — including why PWM, and not a steady voltage, is how information crosses a magnetic gap. A held value would vanish in the core. An edge survives.
Why a ring, specifically? Because a doughnut is a closed magnetic circuit: the flux runs around inside the iron and almost none escapes. That makes it efficient, quiet (little radiated interference), and — crucially for communication — a tight container that a second winding can dip into later. For now, one winding: drive the current, watch the flux circulate the loop, and watch the induced voltage light up only when the current moves.
Switch the drive to steady DC: the flux sits still and the induced voltage flatlines to zero — the toroid has gone blind. Switch back to PWM and every edge fires a voltage spike. That contrast is the whole reason the rest of this series exists.
Pulse-width modulation is almost insultingly simple and quietly everywhere. Take a square wave at a fixed frequency. Make it spend more of each cycle high to mean "more," less to mean "less." The fraction of each period spent high is the duty cycle, and the average of the waveform is exactly D · V — slide the width, slide the value. A cheap low-pass filter on the far end averages the pulses back into a smooth analogue level. That's how a digital chip "outputs" an analogue voltage, dims an LED, or sets a motor's speed — all with a pin that can only ever be fully on or fully off.
The information isn't in how high the pulse goes — it's always the same height. It's in how wide it is. PWM hides a smooth value inside a stream of identical edges.
And identical edges are precisely what a toroid will carry. Hold that thought — it's the join.
Put the two halves together. A toroid passes only change — edges, not levels (§1, §2). PWM hides its message in the timing of edges, not in any level (§3). So PWM is not just a way to drive a toroid — it's the natural way, the one language whose entire content is the thing a toroid can actually carry. You could never send a steady "3.7 volts" through a transformer; the flux would simply walk until the core saturated. But you can send the edges that encode 3.7 volts as a 74%-wide pulse, let them cross the magnetic gap, and rebuild 3.7 volts on the far side — with no wire connecting the two sides at all.
That last part — no wire connecting the two sides — is where this series is headed: communication across a magnetic gap, galvanically isolated, the message riding entirely on change.
Same pattern as before: foundation first, then stack the crossing, the message, and the choke when you're ready.