# how can i calculate the transfer function of this oscillator?

first of all i don't know what is the name of this oscillator,so please tell me its name.

second of all, how to calculate the transfer function for the feedback loop (A B)? simulate this circuit – Schematic created using CircuitLab

• Homework? Have you tried hooking it up or simulating it? I don't believe that's going to work -- it would be a phase shift oscillator if OA1 were connected as an inverting amplifier. However, as shown and assuming no parasitics the thing only has positive feedback at DC -- so the thing will just slam over to one rail or the other and stay there. Jan 25, 2020 at 19:49
• Can you share the original source of this circuit? Are you sure you copied the R & C locations correctly from the original source? Jan 25, 2020 at 19:49
• Do you know how to calculate the transfer function of each stage in the feedback path (i.e. C1, R5 OA4, then C5, R4, OA2, etc)? Jan 25, 2020 at 19:50

The first realization is the condition for sine oscillation with an overall phase shift of 4 stages = 360 deg = 180 + 60+60+60 with unity gain. (See Barkhausen criterion)

The problem with your design is the phase shift is 0+3X=360 X=120 is not possible with a 1st order filter, so it oscillate at the unity GBW product of the Op Amps and the Caps will be AC short circuits and not influence the results, as expected.

Point A will have the near saturation level output.

Then each stage is -6dB @ -180-60=-240 deg= +60

• 3 stages c 60 deg = 180 with an inverter (with 8dB gain) = 360 = 0 deg

So it must be corrected as follows with R2/R1=8.

I'll let you compute the frequency where R/Xc impedance ratio gives 60 deg. simulate this circuit – Schematic created using CircuitLab

Knowing a HPF has 45 deg phase shift at the breakpoint at -3dB you have to compute the impedance ratio where the phase shift is 60 deg a common triangle. Then you can prove what I assert is -6dB @ 60 deg so the gain for the inverter needs to be 18dB which is ~ 8:1 for 1+R2/R1.

I'll let you think about the simple trigonometry.

In reality, the Op Amp gain reduces when it saturates so slightly > 1 loop gain is tolerated with some slight distortion.. Second, in order to initiate a rapid start, the Cap grows slowly and has no full voltage initial condition, so the LPF phase shift oscillator is better than the HPF phase shift oscillator as this one represents. Also the Op Amps need a split supply or else the R's must go to V+/2.