Electronic – 1960’s video showing early computer design & simulation of a low-pass filter circuit

the op-amp is correctly wired up with in an inverting circuit configuration. because of the negative feedback through passive components, the "-" terminal is a virtual ground. the node equations (\$V_2\$ is the voltage at the node where are \$R_1\$, \$R_2\$, \$C_4\$, and \$C_3\$ are connected) are:

$$ \left(\frac{1}{R_1} + \frac{1}{R_2} + sC_4 + sC_3 \right)V_2 - sC_4 V_\text{out} = \frac{1}{R_1} V_\text{in}$$

$$ sC_3 V_2 + \frac{1}{R_5} V_\text{out} = 0 $$

from that, i get

$$ \begin{align} A(s) \triangleq \frac{V_\text{out}}{V_\text{in}} & = \frac{-\frac{1}{R_1} s C_3}{\frac{1}{R_5}\left(\frac{1}{R_1} + \frac{1}{R_2} + sC_4 + sC_3 \right) + (sC_4)(sC_3) } \\ \\ & = \frac{-\frac{1}{R_1 C_4} s}{\frac{1}{R_5 C_3 C_4} \left( \frac{1}{R_1} + \frac{1}{R_2} \right) + \frac{C_4 + C_3}{R_5 C_3 C_4} s + s^2 } \\ \\ & = \frac{-H \omega_0 s}{s^2 + \frac{\omega_0}{Q} s + \omega_0^2} \\ \end{align} $$

equating the corresponding coefficients...

$$ \omega_0^2 = \frac{1}{R_5 C_3 C_4} \left( \frac{1}{R_1} + \frac{1}{R_2} \right) $$

$$ \frac{\omega_0}{Q} = \frac{C_4 + C_3}{R_5 C_3 C_4} $$

$$ H \omega_0 = \frac{1}{R_1 C_4} $$

i think the intent, in the lecture notes posted in the question is that \$ \omega_0 \triangleq 2 \pi f_\text{m} \$

so let \$ C_3 = C_4 \triangleq C \$ and let \$ k \triangleq \omega_0 C \$.

then $$ \frac{1}{R_1} = H \omega_0 C $$ $$ \frac{1}{R_2} = (2Q - H) \omega_0 C $$ $$ \frac{1}{R_5} = \frac{1}{2Q} \omega_0 C $$.

so plug this in for \$R_1\$, \$R_2\$, and \$R_5\$ and see if equality in the three "corresponding coefficients" equations above is met. if so, the transfer function, as given in the question, is correct.

Please remember that when using sim.okawa-denshi.jp it is up to you to know how to properly create the topology in question. It is ill advised to create a Sallen-Key topology with a gain greater than 2 because it tends to oscillate. In fact, the page itself even states (after typing in your numbers) that it will oscillate at a frequency of 401.949Hz. If you require more gain then please create another stage that focuses on that.

Edit: Even better, I noticed that you Q-factor and your Damping factor are both negative. That is a sure indication that you will have oscillatory behavior.

As a suggestion, please use the calculation that asks for fc, gain, and provide either the Q-factor or damping ratio. This will provide you with a more stable set-up (real op amps often require additional components for stability).