Skip to Main Content
A lowpass second-order ChebyÂ¿shev or Butterworth filter is converted into a fourth-order filter with an equiripple or maximally flat passband and a double attenuation pole at a finite frequency using the Zdunek- MÃ¶bius transformation. The double pole enables the filter to be realised as a single lattice with an identical crystal in each arm resonating at the pole frequency. Transferring these crystals to outside the lattice results in filters with a very stable transition band, since the pole frequency now depends only on crystal frequencies, whose variation is several orders of magnitude less than the variation of lumped inductors and capacitors which influence the pole frequency in a conventional crystal filter. The transformation by Zdunek for doubling the circuit order is not applied in the usual manner to obtain a higher-order characteristic of a similar type. Instead, it is used here to obtain a characteristic which is unconventional as a result of the double attenuation pole. This application is not restricted to crystal filters. The symmetrical to asymmetrical transformation is applicable to other type of bandpass crystal filter.