# Why don't infinitely many sinusoidals exist in an oscillator circuit?

A sinusoidal oscillator, takes the ambient thermal noise or any imbalance in the circuit as input and gives out an oscillation.

This kind of oscillators are called to be linear. By the nature of a linear system, if it gives out $Y_1$ when the input is $X_1$ and gives out $Y_2$ when the input is $X_2$, it must give out $Y_1+Y_2$ when its input is $X_1+X_2$.

The thermal noise which starts the oscillations is always present in the system, which means that there are literally infinitely many inputs. Why doesn't it start another oscillation in the system? Why don't other sinusoidal with the same frequency but random phases come out and take place in the system? What physical mechanism prevents existence of a second oscillation and makes the first one dominant?

-
In theory it's random, one phase of the noise is a bit bigger and that grows. In practice, the act of turning it on might be the stimulus that gets it going. I'm not sure any practical oscillator can be truly linear. There has to be some gain control to limit the amplitude. – George Herold Aug 9 '14 at 18:36

A required condition for oscillation in a linear oscillator is that the round trip gain is > 1 (and it will become equal to 1 when oscillation begins and nonlinearities kick in).

Generally we make a narrowband oscillator by making the feedback circuit selective --- it should have low loss for a very narrow band of frequencies, but high loss for other frequencies.

However, no oscillator is perfectly single frequency. There is always some phase noise that results in an output spectrum with some spread around the center frequency. Since frequency is a continuous value, in some sense there are an "infinite number" of frequencies present --- they're just tightly bunched around the center frequency of the oscillator.

-

Quite simply, there is only a single frequency and phase that satisfies the characteristic equation governing the system poles i.e H = -1 or H-1 = 0 where H describes the system poles.

What this means is that the gain must be exactly 1 and the phase must be exactly 180 for a stable oscillation to persist. So this selects just one signal from the wideband signals available in the white noise you're considering.

This explains why one frequency with a particular phase dominates. In reality, uncertainties and jitter result in the selected oscillation frequency being "spread" a little which is known as phase noise. But this is just due to circuit inaccuracies causing a time-variant signal from the oscillator.

-
"...the phase must be exactly 180". Question: Which phase? (Barkhausen criterion requires 0 deg). – LvW Aug 11 '14 at 14:49

In this answer, I like to add something which explains why and how oscillations at one single frequency are possible:

1.) As an alternative to the "system pole" explanation a "linear" oscillator must fulfill Barkhausen´s oscillation condition (which is a necessary one only): Loop gain LG=1 (0 dB).

That means: Unity gain magnitude around the feedback loop and a phase of 360 deg (0 deg). Sometimes, a phase shift of 180 deg is mentioned, however, this requirements neglects the minus sign at the summing node - if it exists (example: phase shift oscillator)! However, there are many oscillators (Wien oscillator) which are based on positive feedback - and in this case only the 360 deg rule applies.

2.) However, Barkhausen´s condition (LG=1) applies to the steady state only (continuous oscillation). Of course, due to tolerances and other uncertainties, this condition cannot be fulfilled exactly. For this reason, and to allow a safe start of oscillations the loop gain is designed from the beginning for a value LG>1 (slighly above unity). Hence, a non-linear amplitude stabilization mechanism is necessary which reduces the loop gain for rising amplitudes (in some simple cases it is the limit set by the power supply). The quality of the oscillation signal primarily is determined by this amplitude stabilization method (diodes, AGC-loop,...)

3.) To ensure that this oscillation condition exists at one single frequency only, the loop gain (not only the feedback circuit!) must fulfill Barkhausen´s condition at one single frequency only. However, it is not necessary that the feedback network is frequency-selektive (bandpass).

There are many counter examples (allpass oscillator, phase shift oscillators, integrator based oscillators, GIC resonators,..). Let´s take as a typical example an allpass based oscillator: The magnitude condition is fulfilled for a broad range of frequency - however, the phase condition (zero resp. 360deg) is fulfilled at one single frequency only. The opposite applies to the integrator-based topologies: The phase condition is met for a broad frequency range but the magnitude condition (0 dB) for one frequency only.

4.) As a summary, I like to add one beautiful sentence which describes the problem of designing a good high-quality oscillator: In order to work as a good "linear" oscillator the circuit must contain a certain degree of non-linearity.

EDIT: As to your question: What physical mechanism prevents existence of a second oscillation and makes the first one dominant?

Your question touches the problem of oscillation start, phase noise and frequency uncertainties. I think, the answer requires some detailed explanations:

1.) At first, I think it is proved that thermal noise plays only a minor role during the starting phase. In reality, it is mainly the power switch-on transients of the reactive network which allows safe start of oscillations - if the condition for the loop gain LG>1 is fulfilled for a frequency in the vicinty of the desired oscillation frequency.

2.) Why in the "vicinity"? For LG>1 the system pole is NOT located at the imaginary axis (as in the ideal case) but slightly shiftet into the right half of the s-plane (RHP). This means: During start of the oscillation the frequency is not exactly as expected.

3.) What happens during and after amplitude/gain limitation? For the sake of simplicity let´s assume that we have a kind of AGC (for example: FET- controlled gain determining resistor). Now - for rising amplitudes, the gain is decreased and the system poles are shifted back to the imag. axis and slightly into the LHP before the control action brings the pole back again - and so forth. Result: Only at the very moment when the poles are directly on the imag. axis the frequency of oscillation assumes its designed value.

4.) Hence, this process of gain control causes the oscillation frequency NOT to be constant. Instead the frequency swings to a certain extent around the desired center frequency. At the same time, also the amplitude does not remain constant. This process (breathing of the amplitude) can be easily observed using circuit simulation. The corresponding time constant is determined by the time constant of the rectifying circuit feeding the controller. As a consequence, the signal output of the oscillator can be seen as a frequency (carrier) which is slightly modulated (FM and AM). However, for a well designed oscillator these deviations are within acceptable limits.

5.) Similar (but not the same ) effects can be observed for amplitude limitations caused by anti-parallel diodes. In this case, we face a rather complicated (non-linear) process consisting of limitation, transient generation, filtering and/or phase shifting. As a result, also in this case the frequency of oscillation slighly varies around the designed value.

-

Two thoughts that come to mind:

1) The sum of two sines at a single frequency is a sine at the same frequency. Therefore if you looked at all the thermal noise, etc which is at or near resonant frequency, you would find only a single sine wave. This is the one which gets amplified.

2) This is a little more far fetched: we know the oscillator amplifies the desired wave until it reaches the correct amplitude and then provides linear gain. Now you asked whether a new sine wave can compete with this one. Consider a new osciallation n sin (wt + theta) - to a constant factor / phase the oscillator sees (sin wt) + n sin (wt + theta) - another sine wave. If the new wave (on the right) where to grow/fall in amplitude (ie if n increased/decreased) then the phase of the sum would be constantly shifting. This (nonlinear) response (with a near constant phase shift toward or away from theta) would look to the linear frequency selective part like an increased or decreased frequency sine wave - this is roughly how some fm modulators work. Therefore we can imagine that such an oscillator must reject large gain in the new wave. This obviously doesn't mean it would have to be less than unity, and phase instability does occur in practice.

That argument wouldn't really apply to smaller gains in n ie slower phase shift of the oscillator, unless the Q of the filer was made unrealisticly large.

My understanding is also that there is no phase preference (unless it is artifically introduced) because the noise is a small signal and sees a linear amplifier and therefore time invariance applies.

-