Passive, active, and digital filters (3ed., CRC, 2009) - tiera.ru

4-30 Passive, Active, and Digital Filtersqav, ð R 1 ÞR 1 ¼ SqR 1jT( jv)j 1R 12(4:156)andqav, ð R 2 Þ jT( jv)jR 2 ¼ SR qR 2þ 12 2(4:157)The derivatives qaqR i(i ¼ 1, 2) are also zero at the points where a ¼ 0 and they are small when a remainsjT( jv)jjT( jv)jsmall [13]. This means that in the passband SR 1 1=2 and SR 2 1=2. Thus, jT( jv)j will sharethe zero sensitivity of a with respect to all the elements inside the two-port, but due to the terms 1=2inEquations 4.156 and 4.157 a change either in R 1 or R 2 will produce a frequency-independent shift (whichcan usually be tolerated) in jT( jv)j in addition to the small effects proportional to qaqR i. This is the basis ofthe low sensitivity of conventional LC-ladder filters and of those active, switched-capacitor, or digitalfilters that are based on LC filter model. This is valid with the condition that the transfer functions of theactive filter and LC prototype are the same and the parameters of the two filters enter their respectivetransfer functions the same way.The Fettweis-Orchard theorem explains why the filtering characteristics sought are those with themaximum number of attenuation zeros (for a given order of the transfer function T(s)). It also helps tounderstand why it is difficult to design a filter that simultaneously meets the requirements of a and w(v)(or to jT( jv)j and b(v)); the degrees of freedom used for optimizing w(v) will not be available to attainthe maximum number of attenuation zeros. It also explains why a cascade realization is more sensitivethan the realization based on LC lossless model. Indeed, assume that, say, one of the characteristics ofFigure 4.3 is realized by two cascaded sections (with the attenuation of each section shown by the dashand-dottedline) with each actual section realized in doubly terminated matched lossless form. Each suchsection of the cascaded filter will be matched at one frequency and the sensitivities to the elements thatare in unmatched sections will be different from zero. In addition, the attenuation ripple in each section isusually much larger than the total ripple, and the derivative qaqx i, which is, in the first approximation,proportional to the attenuation ripple, will not be small. Indeed, practice shows [4] that there is, in fact, asubstantial increase in sensitivity in the factored realization.4.15 Sensitivity of Active RC FiltersThe required component tolerances are very important factors determining the cost of filters. They areespecially important with integrated realizations (where the tolerances are usually higher than in discretetechnology). Also, the active filter realizations commonly require tighter tolerances than LC realizations.Yet two classes of active RC filters have tolerances comparable with those of passive LC filters. These areanalog-computer and gyrator filters that simulate doubly terminated passive LC filters. The tolerancecomparison [4] shows the tolerance advantages (sometimes by an order of magnitude) of the doublyterminated lossless structure as compared to any cascade realization. These are the only methods that arenow used [14] for high-order high-Q sharp cutoff filters with tight tolerances. For less demandingrequirements, cascaded realizations could be used. The main advantages that are put forth in this case arethe ease of design and simplicity of tuning. But even here the tolerance comparison [4] shows that thestages have better tolerances if they are realized using gyrators and computer simulation methods.4.16 Errors in Sensitivity ComparisonsIn conclusion we briefly outline some common errors in sensitivity comparison. More detailed treatmentcan be found in Ref. [4].