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I tried to evaluate the frequency of the following oscillator:

enter image description here

The loop gain \$\beta(j\omega)A(j\omega)\$ is:

enter image description here

To find frequency I must impose the condition:

$$\angle \beta(j\omega_0)A(j\omega_0)=0$$

but I don't know if the term \$1-(RC\omega_0)^2\$ is positive or negative. The angle is different in the two cases.

Thank you for your help.

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  • \$\begingroup\$ Choose the lower \$\omega\$ where it is + (which has more gain ) such that the numerator= demominator and R2/R1 fine tunes that ratio =1.00 for a perfect sinewave if more than 1 then it starts to saturate where gain is automatically reduced. This assumes a split supply otherwise Vin+ shud be Vcc/2 instead of gnd. Essentiall'y 2xRC gives 180 deg lag and R2/R1 amp gives the other 180 inversion for Barhausen Criteria with unity gain \$\endgroup\$ Nov 7, 2016 at 17:56
  • \$\begingroup\$ In your equation a minus sign is missing (gain is -R2/R1). Where is the problem? For w=1/RC the terms [1-(RCwo)²] are equal to zero and the equation reduces to (-R2/R1)x(-1)=+R2/R1 which can be made to be unity. \$\endgroup\$
    – LvW
    Jan 8, 2017 at 10:45

1 Answer 1

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Your circuit is an "allpass oscillator". The transfer function of the open loop is correct, in principle. However, the minus sign of the inverter is missing: -R2/R1.

As you can see, for w=1/T the remaining expression is -(R2/R1)(-2j/2j)=+1. Hence, the oscillation condition is fulfilled at w=1/RC. For a safe start of osillations, the inverter gain should be slightly larger than "-1" (for example: -1.1). The rising oscillation amplitudes will be hard-limited by the power supply rail, unless you include a soft-limiting device (two anti-parallel diodes across R2 - perhaps in series with another resistor).

Explanation: Both blocks are first order allpass filters with a phase shift of -90deg at the frequency w=1/T per unit. Hence, the total phase shift at this frequency is -180deg. The inverter provides another -180deg and the phase condition (360deg resp. 0 deg) of the oscillation criterion is fulfilled at this frequency. The amplitude condition is fulfilled (theoretically) for R2=R1.

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  • \$\begingroup\$ For others benefit, as you know In theory the Vout swing is critically dependent on gain error from 1.00 and even if perfect, matched R's the output would only swing equivalent to the input offset DC voltage. since gain is 1. ... The hard limiting which abruptly reduces gain to 0 thus regulating the output to a full swing with a loop gain of 1 if R ratio is say 1.02. Soft Limiting can also work with more control but also more distortion. One would not choose this for a 60dB SNR sine wave due to tolerances. \$\endgroup\$ Nov 7, 2016 at 20:53

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