Resonant Stub
Stubs are often used as resonant circuits in distributed element filters. An open circuit stub of length will have a capacitive impedance at low frequency when . Above this frequency the impedance is inductive. At precisely the stub presents a short circuit. This is qualitatively the same behaviour as a series resonant circuit. For a lossless line the phase change constant is proportional to frequency,
where v is the velocity of propagation and is constant with frequency for a lossless line. For such a case the resonant frequency is given by,
While this is qualitatively like a lumped element resonant circuit the impedance function is not precisely the same quantitatively. In particular, the impedance will not continue to rise monotonically with frequency after resonance. It will rise until the point where at which point it will be open circuit. After this point (which is actually an anti-resonance point) the impedance will again become capacitive and start to fall. It will continue to fall until at it again presents a short circuit. At this point the filtering action of the stub has totally failed. This response of the stub continues to repeat with increasing frequency alternating between resonance and anti-resonance. It is not only a characteristic of stubs, but of all distributed element filters, that there is some frequency beyond which the filter fails and multiple unwanted passbands are produced.
Similarly, a short circuit stub is an anti-resonator at, that is, it behaves as a parallel resonant circuit, but again fails as is approached.
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