A quick way to identify filters is to check their response at DC (\(f=0\)) and infinite frequency (\(f=\infty\)).
- \textbf{Low-Pass:} Passes DC, blocks high f.
- \textbf{High-Pass:} Blocks DC, passes high f.
- \textbf{Band-Pass:} Blocks both DC and high f.
- \textbf{Band-Reject:} Passes both DC and high f.
Step 1: Analyze the circuit's behavior at very low frequencies (\(f \to 0\)).
At DC or very low frequencies, capacitors act as open circuits.
Capacitor \(C_1\) is in series with the input resistor \(R_1\). Since it's an open circuit, no input signal can reach the op-amp's inverting terminal.
This means the output voltage will be zero. The gain is zero at low frequencies.
Step 2: Analyze the circuit's behavior at very high frequencies (\(f \to \infty\)).
At very high frequencies, capacitors act as short circuits.
Capacitor \(C_2\) is in parallel with the feedback resistor \(R_f\). Acting as a short, it effectively shorts the feedback path, making the feedback impedance zero.
In an inverting op-amp configuration, Gain \( = -Z_f / Z_i \). With \(Z_f \to 0\), the gain of the circuit also approaches zero.
Step 3: Combine the low and high frequency analyses.
Since the circuit has zero gain at both very low and very high frequencies, but will have a non-zero gain at mid-frequencies (where the capacitors have finite reactance), it must be a band-pass filter. It passes a band of frequencies and rejects frequencies that are too low or too high.
