A second order RC circuit frequency response

Not an expert at Latex, neither am one in circuit analysis. My approach involves the use of the concept of transfer functions.

Let $Z_1 = \frac 1{j\omega C} \bigg|\bigg| \left(R+\frac 1{j\omega C}\right)$ , the C-C-R $\pi$ circuit on the right. Then the impedance of the whole RC circuit is $Z = R + Z_1$ . And we have:

$\begin{aligned} \frac {V_{out}}{V_{in}} & = \frac {Z_1}{Z} \times \frac {R}{R+\frac 1{j\omega C}} \\ & = \frac {\color{#3D99F6}Z_1}{R + \color{#3D99F6} Z_1} \times \frac {j\omega CR}{1+j\omega CR} & \small \color{#3D99F6} \text{See note: } Z_1 = \frac {1+j\omega CR}{2j\omega CR - \omega^2 C^2R} \\ & = \frac {\color{#3D99F6}\frac {1+j\omega CR}{2j\omega CR - \omega^2 C^2R}}{R + \color{#3D99F6} \frac {1+j\omega CR}{2j\omega CR - \omega^2 C^2R}} \times \frac {j\omega CR}{1+j\omega CR} \\ & = \frac {1+j\omega CR}{1-\omega^2 C^2R^2 +j\omega CR(1+2R)} \times \frac {j\omega CR}{1+j\omega CR} \\ & = \frac 1{1+2R + j\left(\omega CR- \frac 1{\omega CR}\right)} \end{aligned}$

For the same $|V_{in}|$ , $|V_{out}|$ is maximum , when $\left|1+2R + j\left(\omega CR- \dfrac 1{\omega CR}\right)\right|$ is minimum or

$\begin{aligned} \omega CR - \dfrac 1{\omega CR} & = 0 \\ \omega^2C^2R^2 & = 1 \\ \omega CR & = 1 \\ \implies \omega & = \frac 1{CR} = \frac 1{10^{-6}\times 10^3} = \boxed{1000} \text{ rad/s} \end{aligned}$

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Note:

$\small \begin{aligned} Z_1 & = \frac 1{j\omega C} \bigg| \bigg| \left(R+\frac 1{j\omega C} \right) = \frac {\frac 1{j\omega C} \left(R+\frac 1{j\omega C} \right)}{\frac 1{j\omega C} + R+\frac 1{j\omega C}} = \frac {1+j\omega CR}{2j\omega CR - \omega^2 C^2R} \end{aligned}$

Sweep over the angular frequency and plot the gain at each value of $\omega$ .

$ZC = - \frac{j}{\omega C} \\ Z_1 = R + Z_C \\ Z_2 = \frac{Z_1 \, Z_C}{Z_1 + Z_C} \\ V_1 = V_{in} \frac{Z_2}{R + Z_2} \\ V_{out} = V_1 \frac{R}{R + Z_C} \\ \text{Gain} = \Big| \frac{V_{out}}{V_{in}} \Big|$

The gain has a maximum value of $\frac{1}{3}$ at $\omega = 1000 \text{rad/s}$ . This is not really a low-pass filter since it rejects DC. I will therefore call it a band-pass filter.