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: Examine the circuit's behavior as frequency approaches zero (\(f \to 0\)).At very low frequencies or DC, capacitors behave as open circuits.
\(C_1\), in series with the input resistor \(R_1\), acts as an open circuit, preventing the input signal from reaching the op-amp's inverting terminal. Consequently, the output voltage becomes zero, resulting in zero gain at low frequencies.
Step 2: Examine the circuit's behavior as frequency approaches infinity (\(f \to \infty\)).At very high frequencies, capacitors behave as short circuits.
\(C_2\), in parallel with the feedback resistor \(R_f\), acts as a short circuit, effectively shorting the feedback path and making the feedback impedance zero. Given the inverting op-amp configuration, the gain \( = -Z_f / Z_i \). As \(Z_f \to 0\), the circuit's gain also approaches zero.
Step 3: Synthesize the low and high frequency analyses.Since the circuit exhibits zero gain at both extremely low and extremely high frequencies, but demonstrates non-zero gain at mid-frequencies (where the capacitors possess finite reactance), it functions as a band-pass filter. This type of filter allows a specific range of frequencies to pass through while attenuating frequencies that are either too low or too high.
