The Notch Filter and Band-Stop
At resonance the output dips to 0
Sweep the frequency across a series RLC and the dot on the curve follows the output magnitude at that frequency. This time the output is taken across L and C together, in series, and the impedance of the LC pair is j(ωL − 1/ωC), the difference of the two reactances. At resonance ω_0 = 1/√(LC) that difference becomes 0 and the pair behaves like a short, so the output across it dips deeply to 0. Bring the dot to the notch frequency, where the dashed line stands and the output drops deepest.
Where you tap the output sets the filter
Even with the same series RLC, the kind of filter changes with the element the output is taken across. Taken across the resistor it was a band-pass, passing most strongly at resonance; across the capacitor it is a low-pass, and across the inductor a high-pass. This time the series L and C are treated as one pair and the output is taken across both. The impedance of the LC pair is the inductor's +jωL plus the capacitor's −j/ωC, giving j(ωL − 1/ωC), which changes strongly with frequency.
At resonance the LC is a short
At resonance ω_0 = 1/√(LC), ωL = 1/ωC and the impedance of the LC pair becomes exactly 0: the series LC behaves like a short. Since the output is taken across that short, no voltage appears there, and the output dips deeply to 0 at resonance. This deep valley is the notch. Move away from resonance and the difference of the two reactances grows, the pair's impedance grows, most of the voltage falls across the pair, and the output approaches the input. So a notch filter throws away just the one resonant frequency and passes nearly everything above and below it unchanged.
The complement of the band-pass, and the 4 basic filters
The notch is the exact complement of the band-pass. In the same circuit, the output across the resistor (band-pass) and the output across LC (notch) always add up to the input voltage, and the squared magnitudes of the two gains add to 1 at every frequency. The notch throws away exactly the narrow band the band-pass lets through, which is why notches are used to cleanly remove 50Hz or 60Hz power-line hum or a single interfering frequency. With this, all four basic filters are in place: the low-pass that passes the low end, the high-pass for the high end, the band-pass for a middle band, and the band-stop that throws away one middle point. Choosing where to pass and where to block turned out to be choosing which element to take the output across.
Back to the first screen
As you slid the frequency, the output curve dipped deeply to 0 at resonance, and the valley was at ω/ω_0 = 1, where the dashed line stands. There ωL and 1/ωC became equal, the series LC turned into a short, and the output across the short vanished. Away from resonance the pair's impedance grew and the output came back toward the input. This notch, which throws away just one frequency, is the exact complement of the band-pass taken across the resistor, and together with the low-pass, high-pass and band-pass, all four basic filters are completed simply by choosing where to take the output.