<Dd> H (s) = ω 0 2 s 2 + ω 0 Q ⏟ 2 ζ ω 0 = 2 α s + ω 0 2 (\ displaystyle H (s) = (\ frac (\ omega _ (0) ^ (2)) (s ^ (2) + \ underbrace (\ frac (\ omega _ (0)) (Q)) _ (2 \ zeta \ omega _ (0) = 2 \ alpha) s+ \ omega _ (0) ^ (2))) \,) </Dd> <P> For this system, when Q> ⁄ (i.e., when the system is underdamped), it has two complex conjugate poles that each have a real part of − α . That is, the attenuation parameter α represents the rate of exponential decay of the oscillations (that is, of the output after an impulse) into the system . A higher quality factor implies a lower attenuation rate, and so high - Q systems oscillate for many cycles . For example, high - quality bells have an approximately pure sinusoidal tone for a long time after being struck by a hammer . </P> <Table> <Tr> <Td> <Table> <Tr> <Th> Filter type (2nd order) </Th> <Th> Transfer function </Th> </Tr> <Tr> <Th> Lowpass </Th> <Td> H (s) = ω 0 2 s 2 + ω 0 Q s + ω 0 2 (\ displaystyle H (s) = (\ frac (\ omega _ (0) ^ (2)) (s ^ (2) + (\ frac (\ omega _ (0)) (Q)) s+ \ omega _ (0) ^ (2)))) </Td> </Tr> <Tr> <Th> Bandpass </Th> <Td> H (s) = ω 0 Q s s 2 + ω 0 Q s + ω 0 2 (\ displaystyle H (s) = (\ frac ((\ frac (\ omega _ (0)) (Q)) s) (s ^ (2) + (\ frac (\ omega _ (0)) (Q)) s+ \ omega _ (0) ^ (2)))) </Td> </Tr> <Tr> <Th> Notch (Bandstop) </Th> <Td> H (s) = s 2 + ω 0 2 s 2 + ω 0 Q s + ω 0 2 (\ displaystyle H (s) = (\ frac (s ^ (2) + \ omega _ (0) ^ (2)) (s ^ (2) + (\ frac (\ omega _ (0)) (Q)) s+ \ omega _ (0) ^ (2)))) </Td> </Tr> <Tr> <Th> Highpass </Th> <Td> H (s) = s 2 s 2 + ω 0 Q s + ω 0 2 (\ displaystyle H (s) = (\ frac (s ^ (2)) (s ^ (2) + (\ frac (\ omega _ (0)) (Q)) s+ \ omega _ (0) ^ (2)))) </Td> </Tr> </Table> </Td> </Tr> </Table> <Tr> <Td> <Table> <Tr> <Th> Filter type (2nd order) </Th> <Th> Transfer function </Th> </Tr> <Tr> <Th> Lowpass </Th> <Td> H (s) = ω 0 2 s 2 + ω 0 Q s + ω 0 2 (\ displaystyle H (s) = (\ frac (\ omega _ (0) ^ (2)) (s ^ (2) + (\ frac (\ omega _ (0)) (Q)) s+ \ omega _ (0) ^ (2)))) </Td> </Tr> <Tr> <Th> Bandpass </Th> <Td> H (s) = ω 0 Q s s 2 + ω 0 Q s + ω 0 2 (\ displaystyle H (s) = (\ frac ((\ frac (\ omega _ (0)) (Q)) s) (s ^ (2) + (\ frac (\ omega _ (0)) (Q)) s+ \ omega _ (0) ^ (2)))) </Td> </Tr> <Tr> <Th> Notch (Bandstop) </Th> <Td> H (s) = s 2 + ω 0 2 s 2 + ω 0 Q s + ω 0 2 (\ displaystyle H (s) = (\ frac (s ^ (2) + \ omega _ (0) ^ (2)) (s ^ (2) + (\ frac (\ omega _ (0)) (Q)) s+ \ omega _ (0) ^ (2)))) </Td> </Tr> <Tr> <Th> Highpass </Th> <Td> H (s) = s 2 s 2 + ω 0 Q s + ω 0 2 (\ displaystyle H (s) = (\ frac (s ^ (2)) (s ^ (2) + (\ frac (\ omega _ (0)) (Q)) s+ \ omega _ (0) ^ (2)))) </Td> </Tr> </Table> </Td> </Tr>

Define bandwidth and quality factor of series resonance circuit