Making a Signal: Positive Feedback and Barkhausen

An amplifier that bites its own tail

Climbing into this rung, you arrive carrying one habit of thought you must now deliberately break. All through the amplifier rungs, feedback meant negative feedback — the thermostat forever subtracting a taste of the output from the input to nudge the result back on target, trading raw gain for calm, predictable behaviour. An oscillator is what you get when you turn that idea around and feed a slice of the output back in step with the input, so it reinforces instead of opposing. This is positive feedback, and the startling result is a circuit that needs no input signal at all: it generates its own.

The difference is one of sign. Negative feedback subtracts: push the output up and the loop pulls it back down, which is why it settles. Positive feedback adds: a nudge upward feeds back as a further nudge upward, so the disturbance feeds on itself and grows. You have heard this happen. Hold a microphone too close to the loudspeaker it drives and the system howls — that piercing squeal is an unwanted oscillator, the speaker's sound re-entering the mic, getting amplified, coming back louder, round and round. An oscillator is that howl, tamed and put to work.

But if there is no input, what starts it? The answer is that perfect silence does not exist. Every resistor in the circuit jitters with faint thermal noise — a tiny random voltage stirred up by heat — and that hiss contains a whisper of every frequency. The loop seizes the whisper at the frequency it favours, amplifies it, returns it in step, amplifies it again, and what began as microvolts of noise climbs into a full-grown, steady waveform. The oscillator bootstraps a clean signal out of nothing but the universe's background fuzz.

The Barkhausen criterion: two conditions at one frequency

To pin down exactly when a loop will sustain an oscillation, walk a signal once around it and ask what it looks like when it returns. The amplifier multiplies it by a gain A; the feedback network passes back only a fraction, call it beta. Their product is the loop gain, T = A times beta — the single most important number in the whole subject. (Note this is the raw open-loop trip around the ring, not the tamed closed-loop gain you read off a finished amplifier.) A self-sustaining oscillation demands two things of this loop gain, and they must both hold at the same frequency.

        +--------------------+
   o--->|   amplifier   A    |----+----o  output
   |    +--------------------+    |
   |                             |
   |    +--------------------+    |
   +----|  feedback net  B   |<---+
        +--------------------+
            (B = beta, the fraction fed back)

   loop gain:   T = A times B

   Barkhausen, at the oscillation frequency f0:
     | T | = 1          gain exactly cancels the loss
     phase(T) = 0       (or a full 360 deg) -- feedback in step

Read the two lines plainly. The magnitude condition, the absolute value of T equals 1, says each round trip neither grows nor shrinks the signal: the amplifier puts back exactly what the feedback network bleeds away, so the wave can ride forever at a constant amplitude. The phase condition, a total phase shift of zero (or a whole 360 degrees), says the returning signal arrives perfectly in step with itself — crest landing on crest. That is what makes the feedback positive at this frequency; the two conditions together are the Barkhausen criterion.

Here is the elegant part: the phase condition is usually satisfied at only one frequency. Build the feedback network out of frequency-dependent parts — capacitors, inductors, a resonator — and its phase shift slides with frequency, lining up to zero at just one special value f0. That lone frequency is where the loop oscillates, and choosing the parts is how you design the pitch. The magnitude condition then just sets the gain: if a network feeds back one-third of the signal at f0, the amplifier must supply a gain of 3 to make T = 3 times 1/3 = 1 — and a non-inverting stage with Rf = 20 k and Rin = 10 k delivers exactly 1 + 20/10 = 3.

Start-up and amplitude: living on a knife edge

A loop gain of exactly 1 is a beautiful idea and a practical trap. If T were truly unity from the first instant, an oscillation already running would coast along forever — but the microvolt of noise we need to start things would also be frozen at a microvolt, never growing into a usable signal. Nothing would ever get going. So no real oscillator is built for T = 1 at switch-on.

The fix is to design the small-signal loop gain a little greater than 1 — say T = 1.05, or even 2 or 3 at start-up. Now that whisper of noise grows on every trip around the loop, swelling exponentially out of the hiss. The oscillation builds, breath by breath, exactly as the microphone howl swells from a faint ring to a full scream. The price of starting reliably is that, for a moment, the loop gain is deliberately too high.

1. At switch-on, thermal noise inside the circuit offers a tiny voltage containing every frequency.
2. The feedback network passes the noise back in step only at f0, so only that frequency satisfies the phase condition and survives the loop.
3. Because the loop gain at f0 is greater than 1, that component grows a little on every round trip, climbing exponentially out of the noise.
4. As the amplitude rises, a deliberate nonlinearity quietly lowers the gain until the loop gain settles to exactly 1 — and there the amplitude holds steady.

That last step is amplitude stabilization, and it is what keeps a sine oscillator clean. Left unchecked, a loop gain stuck above 1 drives the amplitude up and up until the wave slams into the supply rails and flat-tops — you get a clipped near-square wave riddled with harmonics, not a pure sine. The cure is a gentle nonlinearity that reduces the gain as the swing grows: a classic Wien bridge uses a small incandescent lamp whose resistance rises as it warms, while other designs lean on diodes or an automatic-gain-control loop. The mechanism nudges T down to precisely 1 and pins the amplitude there.

Two families: the singer and the snapper

Almost every oscillator falls into one of two camps, and the split comes from how it meets the phase condition. The first family is the linear, sine-wave oscillator: drop a sharply frequency-selective network into the loop so the phase lines up at one clean frequency, and the output is a smooth sine wave. At audio frequencies you do it with resistors and capacitors — the Wien bridge and the phase-shift oscillator of guide 2. At radio frequencies you use the resonance of an LC tank, with its high Q sharpening the pitch — the Colpitts and Hartley circuits of guide 3 — and for real precision you replace the tank with a quartz crystal.

The second family does not bother with a resonator at all. A relaxation oscillator charges a capacitor through a resistor — the same RC time constant you metered out in the DC rung — letting the voltage ramp up until it trips a threshold, at which point the circuit snaps to its other state, reverses, and ramps back. The snapping is done by a Schmitt trigger, a comparator with a built-in deadband (its hysteresis, the thermostat-with-a-gap from the comparator rung) so it flips decisively instead of chattering. The output is a square or triangle wave, and its rhythm is set by R and C, not by resonance.

The undisputed champion of the relaxation family is the 555 timer, an eight-pin chip you will meet in guide 4. Wired one way it free-runs as an astable, pumping out a continuous square wave; wired another it becomes a monostable, firing exactly one timed pulse each time you poke it. The fraction of each cycle the output spends high is its duty cycle, and learning to set it independently of the frequency is the heart of pulse control.

Honest notes: jitter, drift, and the oscillator you never wanted

No oscillator holds a perfectly fixed frequency. The very thermal noise that kindled it never leaves, and it now smears each cycle's timing by a hair — random edge-to-edge wobble that goes by jitter in the time domain and phase noise in the frequency domain. On top of that the frequency slowly drifts as temperature, supply voltage, and ageing tug on the components. This is exactly why the crystal oscillator is prized: quartz behaves like an LC resonator with a fantastically high Q, so its frequency barely budges — parts per million, where an RC circuit might wander by percent. Precision is the whole reason guide 3 ends on crystals.

Now the warning that ties this rung back to the last one. Every feedback amplifier you so carefully stabilized is a latent oscillator. If a loop you meant to be negative-feedback picks up enough extra phase shift around its path — from stray capacitance, a capacitive load, too many stages — the returning signal can swing all the way to in-step, and the moment it meets Barkhausen the amplifier breaks into an unwanted song. That is the deep reason you watched phase margin and reached for frequency compensation: keeping an amplifier stable and making an oscillator oscillate are the same physics read from opposite ends. Amplifier stability is just Barkhausen you are trying to avoid.