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I would like to understand how colpitt's oscillator works but I'm confused because there are many configurations on the internet and I don't know what is the difference between them.

For the next two configurations, Why are the capacitors C1 and C2 are moved ? They where between the output ( collector ) and the ground and they became between the output and the Vcc.

http://upload.wikimedia.org/wikipedia/commons/b/bf/NPN_Colpitts_oscillator_collector_coil.svg

http://www.learnabout-electronics.org/Oscillators/images/LC-Colpitts-osc-01.gif

For the following configuration, I understand it very well because the feedback path is very clear.

Would you tell me where the feedback bath of the previous two configurations is ?

http://www.expertsmind.com/CMSImages/1892_colpitts%20oscillator.png

My last questoin is : Why we need to modify the configuaration ? which one is most suitable for radio frequency ? and which one has a stable frequqncy and works without errors ?

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If the active device is a transistor (BJT or FET) there are, in general, three different amplifier configuratios possible: Common emitter, common collector, common base.

The first two references show a common base Colpitt oscillator (base on signal ground). The last ref. contains a Colpitt oscillator in common emitter configuration.

Feedback path:

(a) Common base: The transistor output is connected to an LC resonant block which is high-resistive for the desired frequency. The feedback signal is sampled between both capacitors (cap. divider) and is in phase with the output. This is necessary for positive feedback because in common-base configuration there is no phase shift between emitter and collector. The ratio between both capacitors determines the amount of feedback.

(b) Common emitter: The working principle is somewhat different. The collector is connected to a grounded third order lowpass (r,out-C1 and L-C2). The output of this lowpass has a phase shift of -180deg at the desired frequency. This is necessary because there is a phase shift of another 180deg between base and (correction): collector (in common-emitter configuration). Hence, again we have positive feedback.
(Note that r,out is the transistor`s output resistance at the collector node).

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The feedback in your first two configurations is the junction of the resonating capacitors: C1 and C2 in the first and C2 and C3 in the second. Both feedback the signal to the transistor emitter. This is actually the normal Colpitts configuration. Any basic circuit configuration is subject to variations due to differences in output voltage, output impedance, number of components, power supply requirements, distortion, etc. While all of these circuits are suitable for radio frequencies (the primary application of the Colpitts oscillator), they may not be equal in performance. However, no circuit will work without errors (I'm not sure what you mean by that).

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