Quadrature Oscillator

Quadrature oscillator is an oscillator based on OP. In Figure.1, this topology is combined by a Miller integrator, and noninverting integrator. We usually add a limiter circuit to limit its voltage.  The oscillation condition is based on Barkhausen criterion:

\[L(jw_{0})\doteq A(jw_{0})B(jw_{0})=1\]

In Quadrature oscillator:

\[L(s)= \frac{V_{o2}}{V_{x}}=-\frac{1}{S^{2}C^{2}R^{2}}\]

 \[w_{0}=\frac{1}{CR}\]

The frequency can be changed by change its capacitor and resistor. Therefore, capacitors and resistors are very critical. 

Figure.1 Topology of Quadrature oscillator

In this design, this circuit can output sine and cosine wave in the same time. And I also added two buffer in this circuit to make sure the signal won't be influenced. If the pole is at the right side of s-plane, the output amplitude increase and cause distortion. On the other hand, if the pole is at the left side of s-plane the output amplitude will decrease. 

Noninverting integrator:

The function of Noninverting integrator is delay the wave 90 degree. In this circuit, because the input is sine wave, therefore, it can change the input signal to cosine wave and cause the negative feedback. In this topology, if Rf  is as same as other resistor, the amplitude will decrease. Therefore,  Rf has to small than other resistor. 

Limiter circuit:

Limiter circuit can limit the amplitude of output to make sure the signal won't be distortion.

Miller integrator:

Miller integrator can produce sine wave. The formula:

\[V_{o}(t) = -\frac{1}{CR}\int_{0}^{t}V_{i}dt-V_{o}\]

Buffer:

Buffer, voltage follower, has huge input impedance as a open circuit. The output impedance has very small impedance as a short circuit. The main function is isolate the signal and load to make sure the input signal won't be influenced. 

Summary:
Although this is a very interesting circuit, this design is not used for a long time. The disadvantage is the frequency is  low, and huge capacitor. However, it's still a very good design for me.