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Home Electric Networks Filters Impedance Relations in mDerived Sections  


Impedance Relations in mDerived SectionsThe iterative impedance of a lowpass constantk T section is given by equation 71,_{ }and is
where Z_{K} is the iterative impedance at zero frequency as given by equation 73. For the mderived section of Fig. 42, equation 98 also applies. (See page 193 of reference 5.) The significance of this is that m and a do not appear in equation 98, and hence the midseries iterative impedance of an mderived lowpass filter having any value of m or a is the same as the midseries iterative impedance of the constantk prototype. Thus, a lowpass mderived T section can be connected to its constantk prototype T section without an impedance mismatch occurring at the junction for any frequency. As has been mentioned before, constantk sections have poor iterative impedance characteristics, and mderived sections are used to remedy this. But, the preceding paragraph brings out the point that the midseries iterative impedances for the prototype and the derived sections are the same. Thus, it becomes necessary to investigate the midshunt iterative impedance characteristics of π sections. The midshunt iterative impedance of a lowpass mderived π section is (see page 209 of reference 5)
It is important to note that a, the ratio f_{∞}/f_{c }enters this equation and to recall that a and m are related as shown in equation 96. Thus, the midshunt iterative impedance of an mderived lowpass π filter section varies with the value of a or m, as shown in Fig. 44. From this figure it is seen that if m = 0.6 the midshunt iterative impedance (that is, the iterative impedance of an mderived π section) is substantially constant from zero frequency to cutoff frequency. It also can be proved that the cutoff frequencies of mderived sections and their prototypes are identical. (See page 193 of reference 5.) It will be noted that Figs. 43 and 44 apply both to lowpass and highpass filters. The characteristics of constantk sections, mderived sections, and their uses in composite filters, will now be summarized. (A) Constantk sections have poor iterativeimpedance characteristics as shown by Figs. 25, 28, and 33. (B) Constantk sections do not have sharp cutoff characteristics as shown by Figs. 29 and 34, but they do have high attenuation at frequencies considerably beyond cutoff as indicated by these figures. (C) The midseries iterative impedance of an mderived T section is the same, for any value of m, as the midseries iterative impedance of its constantk prototype; hence, mderived T sections of any value of m or a and constantk prototype T sections can be connected together without reflection loss at the junction. (D) The midshunt iterative impedance of an mderived π section is determined, at any frequency, by the values of m and a; hence, values of m and a may be selected such that very good iterative impedance characteristics can be obtained. (E) The cutoff frequencies of mderived and constantk prototype sections are the same; however, the mderived section has theoretically infinite attenuation at a frequency determined by a and m. Based on these facts, composite filters are constructed as follows: Sufficient constantk T sections are used to give the desired attenuation considerably beyond cutoff; three such sections will provide sufficient attenuation for most purposes. From these prototypes an mderived T section is designed, and this is then "split" into two mderived half sections, which are used to terminate the constantk sections so that
good impedance characteristics and sharp attenuation characteristics are obtained. The mderived half sections are connected so that they offer midshunt terminations (which vary with m and can be made nearly independent of frequency) to networks external to the final composite filter. However, they offer midseries iterative impedances to constantk T sections inside the filter, and hence internal reflections are prevented (Fig. 45).


Home Electric Networks Filters Impedance Relations in mDerived Sections 