@Muse - An inverting active band-pass filter using a single operational amplifier #U1. It combines a high-pass stage at the input with a low-pass stage across the feedback loop.
### Transfer Function and Center Frequency
The complex gain in the frequency domain is given by:
$$A_v(s) = -\frac{Z_f(s)}{Z_{in}(s)} = -\frac{R_2 \parallel \frac{1}{sC_2}}{R_1 + \frac{1}{sC_1}} = -\frac{s R_2 C_1}{(1 + s R_1 C_1)(1 + s R_2 C_2)}$$
The circuit provides:
- **Lower cutoff frequency ($f_L$):** Set by input high-pass elements #R1 and #C1:
$$f_L = \frac{1}{2\pi R_1 C_1} \approx \frac{1}{2\pi \times 1\text{ k}\Omega \times 159\text{ nF}} \approx 1\text{ kHz}$$
- **Upper cutoff frequency ($f_H$):** Set by feedback low-pass elements #R2 and #C2:
$$f_H = \frac{1}{2\pi R_2 C_2} \approx \frac{1}{2\pi \times 10\text{ k}\Omega \times 15.9\text{ nF}} \approx 1\text{ kHz}$$
- **Midband gain ($A_0$):**
$$|A_0| = \frac{R_2}{R_1} = \frac{10\text{ k}\Omega}{1\text{ k}\Omega} = 10 \text{ (20 dB)}$$
With both corner frequencies aligned at $f_0 \approx 1\text{ kHz}$, the filter peaks sharply around the $1\text{ kHz}$ test signal from #Vin, attenuating both DC/low-frequency drift and high-frequency noise while inverting the output phase by $180^\circ$ at resonance.
