operational amplifier - Deriving the formula of oscillation frequency for the Phase Shift Oscillator

My aim is to find the oscillation frequency of a Phase Shift Oscillator.

I start by finding transfer function of the cascaded RC network.

$$V_o(s) = \dfrac{\dfrac{1}{C_3s}}{R_3+\dfrac{1}{C_3s}} V_2(s) = \dfrac{1}{1+R_3C_3s} V_2(s).$$

Similarly,

$$V_2(s) = \dfrac{1}{1+R_2C_2s} V_1(s) \quad\text{and}\quad V_1(s) = \dfrac{1}{1+R_1C_1s} V_i(s).$$

Then the transfer function is:

$$\begin{array}{lcl} H(s) = \dfrac{V_o(s)}{V_i(s)} &=& \dfrac{1}{(1+R_1C_1s)(1+R_2C_2s)(1+R_3C_3s)} \\ &=& \dfrac{1}{R_1R_2R_3C_1C_2C_3s^3 + \dots} \cdots \\ && \dfrac{}{(R_1R_2C_1C_2 + R_2R_3C_2C_3 + R_1R_3C_1C_3)s^2 + \dots} \cdots \\ && \dfrac{}{(R_1C_1 + R_2C_2 + R_3C_3)s + 1} \end{array}$$

So the frequency response is:

$$\begin{array}{lcl} H(j\omega) &=& \dfrac{1}{j\omega \left[ (R_1C_1 + R_2C_2 + R_3C_3) - R_1R_2R_3C_1C_2C_3\omega^2 \right] + \dots} \cdots \\ && \dfrac{}{\left[ 1 - (R_1R_2C_1C_2 + R_2R_3C_2C_3 + R_1R_3C_1C_3)\omega^2 \right]} \end{array}$$

Now, we are looking for a special $\omega$ value, $\omega_0$, for which the argument of $H(wj)$ will be $\pm180^o$. Clearly, it happens when

$$R_1C_1 + R_2C_2 + R_3C_3 = R_1R_2R_3C_1C_2C_3\omega^2 \Big|_{\omega=\omega_0}.$$

Hence we find the oscillation frequency as

$$\begin{array}{rcl} \omega_0 &=& \sqrt{\dfrac{R_1C_1 + R_2C_2 + R_3C_3}{R_1R_2R_3C_1C_2C_3}} \\ \text{f}_0 &=& \dfrac{1}{2\pi} \sqrt{\dfrac{R_1C_1 + R_2C_2 + R_3C_3}{R_1R_2R_3C_1C_2C_3}} \end{array}.$$

When $\quad R_1=R_2=R_3=R\quad$ and $\quad C_1=C_2=C_3=C\quad$:

$$\text{f}_0 = \dfrac{\sqrt{3}}{2 \pi RC}$$

However, according to all online articles including Wikipedia, the formula for the oscillation frequency is

$$\text{f}_0 = \dfrac{1}{2 \pi RC \sqrt{6}}.$$

I did an experiment with the exact circuit I attached above with $R=1k\Omega$, $C=100nF$, $R_i=1k\Omega$ and $R_f=33k\Omega$ by using TL084 opamp. I observed the oscillation period as 7.4ms.

According to the formula I derived above, it should have been

$$\tau_0 = 2 \pi (1k\Omega) (100nF) / \sqrt{3} = 362.76 \text{ns}.$$

And according to the other's formula, it should have been

$$\tau_0 = 2 \pi (1k\Omega) (100nF) \sqrt{6} = 1.539 \text{ms}.$$

Finally, my questions are:

1. Why is the formula I found above is different than the other's formula? Where did I make the mistake? Did using a an extra opamp for buffer affect it?



2. Why does the period of my oscillator differ so much from what the both formula say?