analogfeedbackoscillator
The feedback network used in the Colpitts oscillator is an LC circuit is shown below.
I understand that the phase difference between the terminals 1 and 3 is 180 degrees.
Also the amplifier output is fed at terminal 1 and voltage at terminal 3 is fed back to the
amplifier.
In that case won't the feedback signal will always be in phase with the input of an inverting amplifier?
Then why does the Colpitts oscillator oscillate only at one frequency?
You must discriminate between Colpitt oscillators in common base or common emitter configuration, respectively. And - most important - you must realize that the LC tank contains a capacitive divider circuit.
The mid point between the two capacitors is either grounded (common base) or is connected to the base node (common emitter). In the first case (common base without signal inversion) the LC tank - indeed - produces 360 deg phase shift and this signal is fed back to the emitter.
In the second case (common emitter with signal inversion) the frequency dependent circuitry could be seen as a third order lowpass that consists of an first order RC lowpass followed by an LC lowpass. Such a third-order lowpass produces 180 deg phase shift at the desired frequency. The remaining 180 deg are produced by the inverting characteristic of the common emitter amplifier (between base and collector).
(Comment: The first RC lowpass function is produced with the help of the finite output resistance at the collector node. For opamp realizations (zero output resistance) you must, therefore, add a an additional resistor. Sometimes, This resistor is forgotten in in some papers).
Consider the Wien bridge oscillator circuit: -
It is very similar to the oscillator in the question except that the lower capacitor is is an inductor in the question. At this point alarm bells should possibly start ringing to inform you that replacing the cap with an inductor will not work.
However, lets look at the frequency and phase response of the RC Wien bridge network. This is the filter section of the Wien bridge oscillator circuit (C2 is replaced by L in the question): -
Here's the phase response: -
As can be seen, at resonance, the phase shift is zero as one expects - this produces oscillation at Fr. With C2 changing to an inductor, the phase shift will be 90 degrees and will therefore not oscillate. This last point presumes you understand what happens when L and C are combined at resonance. If not then look on this page for a simulation.
Best Answer
I'm not answering this in the classical "colpitts explanation way" because I think certain things need to be said about the driving source to pin 1 and the off resonance of L and C2 (ignoring C1's effect initially)....
If you apply an oscillating voltage at pin 1, L and C2 form a resonant low pass filter with pin 3 being the output. See below: -
Depending on the Q of the circuit there may be voltage amplification but there will be round about 90 degrees of phase shift between pin 1 and pin 3.
However, this isn't quite how the colpitts works. C1 plays a role and pin 1 isn't a driven by hard voltage source but by a weak voltage source. This adds up to an overall phase shift that can become 180 degrees; the output resistance of the driving source produces a phase shift with C1 that may be (say) 70 degrees and at the right frequency (slightly higher than resonance of L and C2), L and C2 produce another 110 degrees. All this adds up to 180 degrees and the oscillator then oscillates at this frequency.