Electronic – Why does this circuit oscillate

the L and C form an LC tank. Frequency is given by:

The basic frequency-determining circuit is a third-order lowpass consisting of two basic sections (cascade):

• Section 1 (first-order lowpass): r,out-C2 (r,out: dynamic output resistance of gain stage) ,

• Section 2 (second-order lowpass): R4-L1-C3.

The output of section 2 is coupled via C4 into the amplifier input node (finite input resistance r,in). Hence, the frequency of oscillation, which is the frequency that causes a phase shift of -180deg between collector and base node, is determined by all external elements - including r,out and r,in. Therefore, it is a very complicated task to create a formula for the oscillation frequency. This is a typical case for circuit simulation.

UPDATE 1: The calculation by hand is not a simple task because - in addition to the 3rd-order lowpass- there is a 1st-order highpass effect caused by the coupling capacitor C4 which has a surprisingly low value (1nF only).

UPDATE 2 Using a symbol analyzer and replacing the transistor output by an ideal current source (however, with finite input resistance R,in of 8 kOhms) the loop gain expression (frequency-determining part only) is as follows:

Numerator: N(s)=-(C4 L2 Rin) s^2

Denominator D(s)= ( +1) ( + C3 R4 + C4 Rin + C4 R4) s ( + C2 L2 + C4 C3 R4 Rin + C3 L2 + C3 L1 + C4 L2 + C4 L1) s^2 ( + C3 C2 L2 R4 + C4 C2 L2 Rin + C4 C2 L2 R4 + C4 C3 L2 Rin + C4 C3 L1 Rin) s^3 ( + C4 C3 C2 L2 R4 Rin + C3 C2 L1 L2 + C4 C2 L1 L2) s^4 ( + C4 C3 C2 L1 L2 Rin) s^5

It is a 5th order expression because of 5 reactive elments.

If you want you can estimate the influence of the loss resistance R4 - in comparison to all other values. This loop gain function crosses the -360deg line at 81.4 kHz (for R4=0) and at 81.6 kHz (for R4=10 Ohms).

These frequencies seem to be rather realistic if compared with a SPICE simulation based on the real model of the used transistor.

Loop gain phase of 0 deg at f=81.6 kHz (R4=0) and f=82.2 kHz (R4=10 Ohms).

Performing a TRAN analysis in the time domain the circuit was oscillating at f=82.9 kHz (R4=0) and f=83.5 kHz (R4=10 Ohms).

The differences between the small-signal ac analyses and the large-signal Tran analyses are caused by the circuits non-linearities.

UPDATE 3:

Without the influence of L2 (replaced by R2) and neglecting C4 (very large) the classical frequency determining part of the third-order equation for loop gain of the Colpitt oscillator is

N(s) = ( - R2 Rin)

D(s) =( + Rin + R4 + R2) ( + C2 R2 Rin + C2 R2 R4 + C3 R4 Rin + C3 R2 Rin + L1) s ( + C3 C2 R2 R4 Rin + C2 L1 R2 + C3 L1 Rin) s^2 ( + C3 C2 L1 R2 Rin) s^3

In this case the phase cross-over frequencies are 71.2 kHz (R4=0) and 71.3 kHz (R4=10 Ohm). From this result you can derive that your dimensioning causes a relatively large influence of L2 and C4 (normally, to be avoided).

LAST UPDATE:

From the given loop gain functions it is easy to find the expressios for the oscillation frequency: Set s=jw and then set the imaginary part Im[D(jw)]=0.