4. Rectifiers and Filters: From AC to DC

Almost every electronic device runs on DC, but the wall outlet delivers AC. Converting one to the other is the job of the rectifier, and this is the most familiar real-world use of diodes.

4.1 The half-wave rectifier: simplest case

A single diode in series with the load, fed by an AC source:

   AC source ──[D]── + load — GND

When the AC voltage is positive, the diode is forward-biased and current flows. When it is negative, the diode blocks. The load sees only the positive half-cycles, with the negative half-cycles missing.

Mathematically, if the AC source is $v_{in}(t) = V_m \sin(\omega t)$, the output is

$v_{out}(t) = \begin{cases} V_m \sin(\omega t) - 0.7 \text{ V} & \text{if } v_{in}(t) > 0.7 \text{ V} \\ 0 & \text{otherwise} \end{cases}$

Average DC value: $V_{dc} = (V_m - 0.7)/\pi \approx V_m/\pi$. So a 12 V peak AC input gives about 3.8 V DC. Not great — most of the input swing is wasted, and the output is also full of ripple.

Half-wave rectifiers are mostly obsolete; they show up only in very low-power applications (LED night lights, signal demodulation in old AM radios where the audio is recovered by following the envelope).

4.2 The full-wave bridge rectifier: what is in your wall-wart

Use four diodes arranged in a bridge:

        AC1 ────┬─[D1]─┐
                 │      │
                [D3]    │
                 │      │
                GND    + out
                 │      │
                [D4]    │
                 │      │
        AC2 ────┴─[D2]─┘

Trace current paths: when AC1 is positive (and AC2 negative), the path is AC1 → D1 → load → D2 → AC2. When AC1 is negative (and AC2 positive), the path is AC2 → D3 → load → D4 → AC1. Current flows through the load in the same direction either way. Both half-cycles deliver power to the load.

The output is now a series of positive humps — the absolute value of the AC waveform — at twice the line frequency (100 Hz from a 50 Hz line, or 120 Hz from 60 Hz).

Average DC value: $V_{dc} = 2(V_m - 1.4)/\pi \approx 2V_m/\pi$ (subtract two diode drops, since two diodes are in series in any current path). A 12 V peak gives about 6.8 V DC.

The bridge rectifier is the workhorse: every wall-wart, laptop charger, phone charger has one. Open any wall-wart and you will find four diodes (or one packaged "bridge rectifier" module that is just four diodes in a single epoxy block).

4.3 The center-tapped full-wave rectifier: historical alternative

Uses a transformer with a center-tapped secondary and only two diodes:

   transformer secondary:
        ─ ┬ ─        AC1 (top)
          │
          ├── center tap (GND for output)
          │
        ─ ┴ ─        AC2 (bottom)
 
   AC1 ─[D1]─┐
             │── + out
   AC2 ─[D2]─┘

Two diodes instead of four, but you need the center-tapped transformer, which is more expensive. Mostly obsolete; the bridge wins.

4.4 Filtering: smoothing the lumpy DC

The raw output of a rectifier is pulsing DC. We want flat DC. The standard solution is a big filter capacitor in parallel with the load:

   [bridge rectifier]── + ──┬────── load
                            │
                            C (filter)
                            │
                           GND

The cap charges up to near the peak of the rectified waveform. Between peaks, when the rectifier output dips, the cap discharges through the load, holding the voltage up. The result is a mostly-flat DC with a small ripple voltage superimposed.

The ripple voltage (peak-to-peak) is approximately:

$V_{ripple} \approx \frac{I_{load}}{f \cdot C}$

where $f$ is the frequency of the rectifier output (100 or 120 Hz for full-wave from 50/60 Hz mains). A 1 A load with a 1000 µF cap on a 120 Hz rectified signal gives about 8 V of ripple. That is a lot. We typically want < 100 mV ripple, so we need much bigger caps — 10 mF or more for high-current supplies — followed by a regulator chip (linear or switching) to clean up the rest.

Bucket-and-pump analogy. Imagine a leaky bucket slowly draining (the load) being refilled twice per cycle by a fire hose (the rectifier pulses). If the bucket is small, the level changes a lot between refills. Make the bucket bigger, the level barely changes — you have a steady output. The cap is the bucket; its size is its capacitance.

4.5 Better filters: LC and π

For lower ripple, you can add an inductor in series after the cap, then another cap in parallel — an "LC filter" or "π filter":

   ── + ──┬──[L]──┬── filtered + ──
          │       │
          C1      C2
          │       │
         GND     GND

The inductor presents a high impedance to AC ripple but low impedance to DC; the second cap continues filtering. Used in audio amplifiers and lab supplies where ripple matters and bulk passive filtering is cheaper than a switching regulator.

Modern reality: for almost any modern device, the rectifier+capacitor is followed by a switching regulator (boost, buck, flyback) or a linear regulator (LDO, 78xx series). These do the heavy lifting of voltage regulation, leaving the rectifier+cap to simply convert AC to roughly-flat DC.

4.6 Specifications you will see on diode datasheets

When picking a rectifier for a real circuit:

• Average forward current $I_F$. What the diode can carry continuously. A 1N4148 (signal): 200 mA. A 1N4007 (1 A rectifier): 1 A.
• Peak inverse voltage (PIV) or $V_R$. Maximum reverse voltage the diode can withstand. For a bridge-rectified 12 V transformer, you want PIV at least $V_m \approx 17$ V and ideally 50 V or more for margin. The 1N4007 has 1000 V PIV — wildly overkill for most uses, but it is cheap and stocked everywhere.
• Forward voltage at rated current — typically 0.7 V at 1 A for silicon, 0.3 V for Schottky.
• Reverse recovery time. Important if you are switching at 100 kHz or higher.