Low pass, high pass composite filters.
The plots shown of image impedance, attenuation and phase change are the plots of a low-pass prototype filter section. The prototype has a cut-off frequency of ωc = 1 rad/s and a nominal impedance R0 = 1 Ω. This is produced by a filter half-section where L = 1 henry and C = 1 farad. This prototype can be impedance scaled and frequency scaled to the desired values. The low-pass prototype can also be transformed into high-pass, band-pass or band-stop types by application of suitable frequency transformations.
Several L half-sections may be cascaded to form a composite filter. Like impedance must always face like in these combinations. There are therefore two circuits that can be formed with two identical L half-sections. Where ZiT faces ZiT, the section is called a Π section. Where ZiΠ faces ZiΠ the section formed is a T section. Further additions of half-sections to either of these forms a ladder network which may start and end with series or shunt elements
It should be born in mind that the characteristics of the filter predicted by the image method are only accurate if the section is terminated with its image impedance. This is usually not true of the sections at either end which are usually terminated with a fixed resistance. The further the section is from the end of the filter, the more accurate the prediction will become since the effects of the terminating impedances are masked by the intervening sections. It is usual to provide half half-sections at the ends of the filter with m = 0.6 as this value gives the flattest Zi in the passband and hence the best match in to a resistive termination.
A crystal filter is a special form of quartz crystal used in electronics systems, in particular communications devices. It provides a very precisely defined centre frequency and very steep bandpass characteristics, that is a very high Q factor—far higher than can be obtained with conventional lumped circuits.
A crystal filter is very often found in the intermediate frequency (IF) stages of high-quality radio receivers. Cheaper sets may use ceramic filters (which also exploit the piezoelectric effect), or tuned LC circuits. The use of a fixed IF stage frequency allows a crystal filter to be used because it has a very precise fixed frequency.
The most common use of crystal filters, is at frequencies of 9 MHz or 10.7 MHz to provide selectivity in communications receivers, or at higher frequencies as a roofing filter in receivers using up-conversion. Ceramic filters tend to be used at 10.7 MHz to provide selectivity in broadcast FM receivers, or at a lower frequency (455 kHz) as the second intermediate frequency filters in a communication receiver. Ceramic filters at 455 kHz can achieve similar bandwidths to crystal filters at 10.7 MHz.