The Notch Filter and Band-Stop
At resonance the output sinks to zero
Sweep the frequency across a series RLC. This time take the output across the series L and C together. The impedance of the LC pair is j(ωL − 1/ωC), the difference of the two reactances. At the resonance ω_0 = 1/√(LC) that difference is zero, so the LC acts as a short, and the output across it sinks deeply to zero. Find the notch frequency where the output drops the deepest.
Where you tap the output sets the filter
In the same series RLC, the kind of filter depends on which element you take the output across. Across the resistor it was a bandpass, peaking at resonance. Across the capacitor it is a low-pass, across the inductor a high-pass. This time we treat the series L and C as one pair and take the output across it. The impedance of the LC pair is the inductor’s +jωL plus the capacitor’s −j/ωC, that is j(ωL − 1/ωC), which varies strongly with frequency.
At resonance the LC is a short
At the resonance ω_0 = 1/√(LC), ωL = 1/ωC and the impedance of the LC pair is exactly zero. The series LC acts like a short. Taking the output across that short, no voltage stands across it, so the output sinks deeply to zero at resonance. This deep trough is the notch. Away from resonance the difference of the two reactances grows, the LC impedance grows, most of the voltage stands across the LC, and the output nears the input. So the notch filter picks out the single resonant frequency to throw away and passes nearly all of the rest above and below.
The complement of the bandpass, and the four filters
The notch is the exact complement of the bandpass. In the same circuit, the output across the resistor (bandpass) plus the output across the LC (notch) always equals the input voltage, so the squared magnitudes of the two gains add to one at every frequency. The narrow band the bandpass passes, the notch throws away exactly. That is why a notch is used to cleanly remove a single frequency such as 60 Hz mains hum or a particular interferer. With this, the four basic filters are all in hand: the low-pass that passes the low side, the high-pass for the high, the bandpass for a middle band, and the band-stop that discards a single middle point. Choosing what to pass and what to block was, all along, choosing which element to take the output across.
Back to the first screen
As you slid the frequency, the output curve sank deeply to zero at resonance, its trough at ω/ω_0 = 1. 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 LC impedance grew and the output recovered toward the input. This notch, picking out the single frequency to throw away, is the exact complement of the bandpass taken across the resistor. Together with the low-pass, the high-pass, and the bandpass, merely choosing where to tap the output completes all four basic filters.