From RC to a tank: the LC resonator
The Wien bridge and phase-shift oscillators of guide 2 are lovely up to a few hundred kHz, but each RC stage only nudges the phase along gently, and the frequency wanders as the parts drift. Climb toward radio frequencies and we want a part that naturally rings at one frequency instead of being coaxed there. That part is the LC tank: an inductor wired across a capacitor. The cap is a bucket that fights any sudden change in voltage; the inductor is a flywheel that fights any sudden change in current. Charge the cap, let it dump its charge into the coil so current builds, then the coil's collapsing magnetic field shoves the charge back into the cap the other way round — the energy sloshes back and forth all by itself.
That sloshing is resonance, and it happens at one clean frequency: f = 1 / (2 times pi times sqrt(L times C)). Pick L = 10 uH and C = 100 pF and you land at f = 1 / (2 times pi times sqrt(10 times 10^-6 times 100 times 10^-12)) which is about 5 MHz — right where RC oscillators get clumsy. A real tank loses a little energy to resistance every cycle, so left alone it rings down and dies, exactly like a pushed swing slowly stopping. The whole job of an LC oscillator is to top up that lost sip of energy on every cycle, at the right moment, just like giving the swing a gentle push each time it comes back.
Colpitts and Hartley: tap the tank for feedback
Recall the Barkhausen criterion from guide 1: for steady oscillation the signal must travel once around the loop and come back the same size (loop gain magnitude exactly 1) and in phase (total phase 0, i.e. 360 degrees). An LC oscillator is just an amplifier whose load is the tank, with a slice of the tank's swing fed back to the input as positive feedback. The tank does double duty: it sets the frequency, because only at resonance is the phase right, and it provides the feedback fraction through a divider. There are two natural ways to build that divider — and they give the two classic topologies.
The Colpitts splits the capacitor into two in series, C1 and C2, and taps the junction between them. The Hartley instead splits the inductor (or uses a centre-tapped coil) and taps the middle. Either way the amplifier supplies roughly 180 degrees of phase and the tank's divider supplies the other 180 at resonance, summing to the full 360 the criterion demands. The feedback fraction is set by the divider ratio: in a Colpitts it is roughly beta = C1 / C2, so the amplifier must deliver a gain of at least C2 / C1 to push the loop gain to 1.
COLPITTS (capacitive divider taps the tank)
+---[ L ]---+----o Vout (full tank swing)
| |
| === C1
| |
| +----o feedback tap -> amp input
| |
| === C2
| |
+-----------+----o GND
|
( amp restores the lost energy each cycle )
series cap Cs = C1 x C2 / ( C1 + C2 )
frequency f = 1 / ( 2 x pi x sqrt( L x Cs ) )
feedback beta ~ C1 / C2 -> amp gain must be >= C2 / C1
HARTLEY = same idea, but split the INDUCTOR
(tapped coil) instead of the capacitor.Put numbers on the Colpitts. With C1 = 100 pF and C2 = 1000 pF the series value is Cs = (100 times 1000) / (100 + 1000) which is about 91 pF; pair that with L = 10 uH and f = 1 / (2 times pi times sqrt(10 times 10^-6 times 91 times 10^-12)) is about 5.3 MHz. The feedback fraction is C1 / C2 = 0.1, so the transistor needs a voltage gain of at least 10 to start. A handy memory hook: Colpitts has the Capacitors, Hartley has the... coil. In practice the Colpitts wins most modern designs, because two small, stable capacitors are cheaper and behave better than a tapped inductor, and there is less stray inductance to misbehave.
Start-up, amplitude, and where LC drifts
There is the same start-up paradox from guide 1. To get going from nothing the loop gain must be greater than 1 so the faint noise already in the circuit grows; but to settle at a constant amplitude it must fall to exactly 1, or the swing would grow forever. The cheap and common fix is to design the gain a few times over 1, then let the swing grow until the transistor runs out of headroom — it starts to clip, and its effective gain droops as the peaks near the supply rails. That nonlinearity automatically drags the average loop gain back down to 1. It works, but the clipping injects harmonics; mercifully the high-Q tank filters most of them away, so the output stays a respectable sine. Cleaner designs add a gentle amplitude-stabilization loop instead, exactly as the Wien bridge did in guide 2.
Now the honest limit of an LC oscillator: its frequency is only as steady as L and C, and both drift. Capacitors and coils change value with temperature; the transistor's own junction capacitances sneak into the tank and shift with bias and heat; even a nearby hand or piece of metal nudges it. So an everyday LC oscillator holds its frequency to roughly 0.1 to 1 percent — that is 1000 to 10000 ppm of wander. Perfectly fine for the tunable local oscillator in a radio, and hopeless for anything that has to keep time. You can even lean into this softness deliberately by replacing part of the tank capacitance with a varactor — a diode whose capacitance changes with reverse voltage — so a control voltage tunes the frequency. That is the seed of the voltage-controlled oscillator you will meet in guide 5.
The crystal: a quartz resonator with astonishing Q
If you want a thousandfold better stability, throw away the LC tank and ring a sliver of quartz instead. A quartz crystal is piezoelectric: squeeze it and it produces a voltage; apply a voltage and it physically flexes. A precisely cut sliver vibrates mechanically at a sharp natural frequency, like a microscopic tuning fork. Because it is a clean mechanical resonance rather than a leaky electrical one, its Q is enormous — tens of thousands to over a million, versus the couple of hundred an LC tank could ever manage.
Electrically, the crystal behaves like an LC tank with a startling equivalent circuit: a series 'motional' arm of a huge effective inductance, a tiny series capacitance, and a small resistance, all sitting in parallel with the package and electrode capacitance C0. The motional values are physically absurd — henries of inductance and femtofarads of capacitance — numbers you could never reach with a real coil and cap, and exactly why its Q and steadiness leave LC in the dust. One subtlety: a crystal has two close resonances, a lower series resonance and a slightly higher parallel resonance, and a given circuit is designed to use one or the other.
Put precision in numbers. A plain crystal holds its frequency to about 10 to 50 ppm (parts per million); 10 ppm at 10 MHz is a wander of just 100 Hz. Add temperature compensation (a TCXO) and you reach roughly 1 ppm; put the crystal in a tiny temperature-controlled oven (an OCXO) and you hit around 0.01 ppm. And the famous 32.768 kHz watch crystal is chosen for one delicious reason: 32768 = 2^15, so a 15-stage binary divider turns it into an exact 1 Hz tick — the heartbeat of every real-time clock.
Crystal oscillator circuits, jitter, and choosing
The everyday crystal oscillator is the Pierce circuit: a single inverting gate — a CMOS inverter, or the built-in inverter inside a microcontroller — with the crystal connected from output back to input and two small load capacitors, one from each end to ground. It is essentially a Colpitts in which the crystal plays the role of the inductor. Because the quartz is so ferociously frequency-selective, it forces the loop to oscillate at its rated frequency and nowhere else, no matter what the lazy gate would have done on its own.
The one spec you must honour is the load capacitance. A parallel-resonant crystal is cut to hit its marked frequency only when it sees a specified load cap, often 12 to 20 pF; the two load caps plus the stray capacitance must add up to that value, or the part sits a few ppm off and your clock is subtly wrong. A few honest gotchas: drive the crystal too hard and you age it or even crack it; stray capacitance on a breadboard throws the load off and can stop it ever starting; and although the frequency is rock-steady, the short-term phase noise — its jitter, the tiny cycle-to-cycle timing wobble — still matters in fast digital links and RF, and is set largely by the circuit around the crystal, not the crystal alone.
So how do you choose? Reach for an LC oscillator — a Colpitts or Hartley — when you need a high or tunable frequency and percent-level accuracy is plenty: radio front ends and tunable local oscillators. Reach for a crystal whenever the frequency must simply be exact and stable: microcontroller and real-time clocks, USB, and the reference inside a radio. And when you need a frequency that is both tunable and locked to a crystal's accuracy, you marry a voltage-controlled oscillator to a crystal reference inside a phase-locked loop — which is exactly where guide 5 takes you next.