## Sunday, July 13, 2014

### The condition of producing oscillation in oscillator

To make an oscillator requires amplifiers or Opamp. The input signal to opamp can be connected in non-inverting and inverting terminal. A non-inverting amplifier produces no phase shift between the signal at the input and output. But an inverting amplifier produces a phase shift of 180 degree.

When an inverting operational amplifier is used as an oscillator, the feedback network must produce a phase shift of 180 degree so that the feed backed signal is again in phase with the original input signal. This ensures positive feedback as required for the oscillation.

The figure below shows a circuit drawn in Matlab 2014 and in the figure the input signal Vs is connected to the negative terminal of the opamp so the signal Vo coming out operational amplifier is has a phase which is 180 degree away from the input signal Vs.

The feedback network provides adds a phase shift of 180 degree to the signal Vo and the ouput signal Vf is then in phase with the signal Vs.

The condition for the sustained oscillation is that the feed back signal Vf should act as Vs. In other words, the amplifier pluse the feedback network should produce signal with only Vf without Vs.

Now,
$A=\frac{V_o}{V_{in}}$   ----(1)
therefore,
$V_o=AV_{in}$      ----(2)
Similarly, Vf is,
$V_f=\beta V_{o}$   ---(3)
and placing Vo from (2) into (3) gives,
$V_f=A\beta V_{in}$    ---(4)
And the oscillation should happen with the input signal Vs when Vs=Vf which gives,
$|A\beta|=1$    ----(5)

So in order to produce sustained oscillation without the external signal we require-
(1) that the total phase shift around the loop should be 0 or 360 degree
(2) the product of open loop gain and beta should be 1 as per the equation (5)

See the earlier post oscillator in communication system