I am just into oscillators where I learned \$AB=1\$ for sustaining oscillations in positive feedback. Since \$A\$ and \$B\$ are both frequency-dependent, \$AB=1\$ is true only for a particular frequency.
What happens to those frequencies for which \$AB>1\$ holds??
Will these frequencies keep on getting amplified until the limiter circuit limits them?
Then why don't we get those frequencies in our output??
Why do we get only one frequency as output in oscillators?
Oscillators work at one frequency by ensuring two things: -
So, if we design a phase-shifting network that has a unique phase-shift for each frequency it handles, we will get an oscillator but, only if the signal fed back is sufficient in amplitude to sustain oscillation.
However, some phase shifting networks can produce a phase shift that is a multiple of the basic oscillation frequency. In other words if 1 MHz produces a phase shift of 360 degrees, maybe some higher-frequency might produce 720 degrees (2 x 360). This could potentially give rise to a sustained oscillation at two frequencies (usually deemed undesireable).
So, we design the phase-shifting network to ensure that the higher-frequency "in-phase" candidate is much lower in amplitude than the "basic" candidate and, given that we only allow the gain to be unity or slightly higher (to accommodate losses in the phase shift network) for the frequency we want, the higher-frequency candidate will not cause oscillation.
The above is also referred to as the Barkhausen criteria.
So what happens to those frequencies which have AB>1??
Saturation.
Let's say there are several frequencies with loop gain \$AB \ge 1\$ and \$n2\pi\$ phase shift, but let's call the one with the highest loop gain \$f_x\$. For \$f_x\$, \$AB > 1\$ and you might expect it to produce an oscillation with amplitude increasing in time. But no real circuit can have its output increase in amplitude indefinitely. There is usually some saturation behavior that limits the output amplitude.
And when this happens, it tends to reduce the gain for all frequencies, not just the one that had the super-unity loop gain. So accounting for saturation, this frequency \$f_x\$ will end up with \$AB=1\$ and all the other frequencies that linear analysis told you had \$AB \ge 1\$ but less than at \$f_x\$, now have \$AB < 1\$, so they no longer oscillate indefinitely.
A short answer from my side:
You must not think in magnitude terms only. Don`t forget the phase. The product AB must be a REAL one . A frequency-selective circuitry has a magnitude as well as a phase which is a function of frequency. And - for a correct design - there will be only one single frequency which can fulfill both conditions at the same time (Barkhausens oscillation criterion with loop gain AB=1):
|A*B|=1 (for practical reasons somewhat larger than "1", for example "1.2") and
phaseshift exp(j*phi)=1 (phi=0).
For this purpose, most known oscillators use lowpass, highpass or bandpass filters as feedback elements. But there are also other (more advanced) topologies.
When saturation occurs, the loop gain (GH or AB) drops to zero, except during the linear transition of the output. The crystal acts as a bandpass filter to produce a sine wave at the input which may also contain harmonics, but the slew rate of the square wave output is generally much faster than the sine wave input, so the harmonic energy has insufficient outline linear time to amplify when it is not saturated and the gain is zero, thus suppressed.
However, for smaller crystal structures >= 33 MHz resonance the gain of the harmonics tends to be higher than the fundamental. Thus you will find these classified as "overtone crystals".
For CMOS feedback oscillators, often a series R (3 kΩ ~ 10 kΩ) from the output is used to limited uW power dissipation in microslice crystals AND in high frequency >> 10 MHz also create additional attenuation of harmonics from the RC effects with the first load capacitor. The most common is third harmonic or "overtone", but higher overtones are used >> 150 MHz.
But when selective harmonics are desired for oscillation (3, 5, 7, etc.) then either how the crystal is processed or additional passive LC tuning helps to boost the harmonic of choice.
The most common warning for XO designs "Never use a buffered inverter" (three linear gain stages vs. one) to avoid amplification of spurious harmonics. When they saturate the inverter and the gain drops to zero, they suppress the fundamental frequency except for a short transition interval. They can behave like an injection locked loop (ILL) where it may randomly oscillate at the fundamental or harmonic depending relative gains and startup conditions. But with a buffered inverter there is more chance during the output transition time to cause spurious harmonic glitches on the transitions and lock onto harmonics.
However, those who successfully used a buffered inverter (myself included) for an XO can now understand that the type of crystal and relative lower gain of the harmonic protected the XO from locking onto the desired fundamental frequency. In some cases, this can be an advantage, but that's a different question.
The Barkhausen stability criterion says: \$|A\ \beta| = 1\$ and \$\angle A\ \beta = 0\$.
Where \$A\$ = amplifier gain,
and \$\beta\$ = feedback attenuation.
So if \$|A\ \beta| = 1\$, then the oscillator will be stable. The feedback loop feeds a portion of the output \$v_o\$ back to the input, \$v_f\$. The amplifier amplifies the input \$v_f\$ to make a larger output, \$v_o\$.
If \$|A\ \beta| > 1\$, then the oscillator will drive itself into saturation and clip the output waveform. The amplifier is an operational amplifier with \$\pm\$ power rails. The amplifier cannot drive the output past the power rails.
The gain and attenuation are not stable and the amplifier output increases to the power rails of the amplifier. If it is a sine wave oscillator, the output increases until the amplifier saturates, and it is no longer a sine wave. Tops get clipped.
If \$|A\ \beta| < 1\$ the oscillations will fade out. This is called damping.
Given that: an oscillator circuit is designed to oscillate at a fixed frequency if \$|A\ \beta| = 1\$ and \$\angle A\ \beta = 0\$ (phase angle is 0\$^\circ\$).
So the crux of your question is: Why don't oscillators oscillate at other frequencies? This is governed by the components used (resistors, capacitors, inductors, and amplifiers).
Although all answers are correct, I believe these are all missing the spirit of your question.
The term "oscillator" generally applies to a circuit specifically designed to produce an AC waveform at a specific frequency. This entails some design choices intended to minimize unwanted effects. This is particularly true for linear oscillators (which is the loop-gain case stated in your question).
You specifically design the gain to be slightly larger than 1 at a specific frequency and you design/rely on non-linearities in the system to keep the oscillation stable. If you allow the gain to be much larger than 1 then you stop having a linear oscillator.
However, this useful engineering simplification comes from having the loop gain only be slightly larger than one which allows you to treat it as a linear oscillator, when in reality it is not. What you actually have is the simplified border case of a non-linear dynamical system with a stable periodic orbit that approaches a sinusoid.
If you further develop that dynamical system (for example by making AB>>1) you can reach another extreme, a very non-linear but stable relaxation oscillator or in intermediate cases you will find a period doubling sequence that creates a chaotic oscillator such as Chua's circuit or a Van Der Pol oscillator.
This image is from an implementation of Chua's circuit you can see that it behaves somewhat as a combination relaxation oscillator/linear oscillator. But the "relaxation component" is non-periodic and long-term unpredictable.
There are uses for all of those alternatives, but linear oscillator theory specifically stays away from those conditions.
