# Maths behind a colpitt oscillator tank circuit

Discussion in 'General Electronics Chat' started by silv3r.m00n, Mar 8, 2012.

1. ### silv3r.m00n Thread Starter New Member

Apr 15, 2010
14
0
Hello

This is a typical feedback circuit of a colpitt oscillator (with inverting amplifier) :

The amplifier output (say the collector) is put into Vin and the feedback is taken from Vout and fed into amplifier input (say the base).

Before I ask my question I would like to put forward some of my observations :

1. XL + Xc2 is in parallel to Xc1 , so working out the complex impedence when XL = Xc1 + Xc2 at F(resonant) the impedence is infinite and the circuit oscillates.

I derive it as : (jXL - jXc2 * -jXc1) / j(XL - Xc1 - Xc2)
For impedence to be infinite => denominator = 0 => total inductive impedence = total capacitive impedence

2. When circuit oscillates , the current in both first and second branch is exactly equal and opposite at any given instant of time. Due to this the voltage across Xc2 is just opposite (or 180deg out of phase) with the voltage across Xc1(which is the input voltage).

So Vout is 180deg out of phase with Vin ( with respect to ground) . This along with an inverting amplifier satisfies the criteria for oscillation.

Also the value of capacitor C2 compared to C1 controls the magnitude of the voltage across it. Smaller the value of C2 , higher is the Vout and higher is the feedback.

Correct me if any of my above assumptions are wrong.

Now my question is :

1. It appears to me that (ignoring Xc1) at any frequency as long as XL > Xc2 , Vout will be 180deg out of phase from Vin

I try to derive it like this :

Vout = Vin * -jXc2 / j(XL - Xc2) = Vin * Xc2(-90) / (XL - Xc2)(90) = Vin * Xc2(-180)/(XL - Xc2)

In that case the circuit should produce oscillations for frequencies > Fr (where XL > Xc2) ?

But that does not happen. It oscillates only at Fr

Is it because at higher frequencies the impedence Xc2 falls and hence the feedback amount also falls making those frequencies unable to grow , but even in that case for frequencies slightly higher than Fr should be able to grow to significant levels and show in output ?

Regards
Silver

Last edited: Mar 9, 2012
2. ### cowades New Member

Feb 6, 2012
9
1
You are correct that Xc dropping off and Xl increasing is the cause (Vo dropping off rapidly above Fr). Feed back is the answer to the rest of the question (Vo=-Vin). Maybe if you think of it as forcing the gain to unity at Fr so higher frequencies don't meet the gain criteria to oscillate.

3. ### Brownout Well-Known Member

Jan 10, 2012
2,375
998
That looks right for perfect L's and C's. You might even get it to work on a simulator by setting Q=infinity, until it blows up that is. But it won't work for finite Q, and so C1 is necessary to achieve 180 degree phase.