Question

what type of filter is this?
passive band - reject
passive low - pass
passive band - pass
active low - pass

Explanation:

• Passive vs Active: The circuit has only passive components (resistor \(R\), capacitor \(C\), inductor \(L\)) with no active components like op - amps. So it is a passive filter.
• Band - reject characteristics: A band - reject filter (also known as a notch filter) is designed to attenuate a specific range of frequencies (the "band" to be rejected). In an \(RLC\) series circuit, the impedance \(Z=\sqrt{R^{2}+(X_{L} - X_{C})^{2}}\), where \(X_{L}=\omega L\) and \(X_{C}=\frac{1}{\omega C}\). At the resonant frequency \(\omega_{0}=\frac{1}{\sqrt{LC}}\), the impedance of the \(LC\) combination is minimum (\(X_{L}=X_{C}\), so \(Z = R\)). But for frequencies around \(\omega_{0}\), the output voltage \(v_{out}\) (across \(LC\)) is relatively low compared to frequencies far from \(\omega_{0}\). A low - pass filter would pass low frequencies (\(v_{out}\) proportional to \(X_{C}\) for low frequencies in a series \(RC\) - like behavior, but here we have \(L\) in series). A band - pass filter would pass a range of frequencies (not the case here as the output is taken across \(LC\) which attenuates around \(\omega_{0}\)).

Answer:

Passive band - reject