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Impedance Relations in m-Derived Sections

The iterative impedance of a low-pass constant-k T section is given by equation 71, and is

where ZK is the iterative impedance at zero frequency as given by equation 73. For the m-derived 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 m-derived low-pass filter having any value of m or a is the same as the mid-series iterative impedance of the constant-k prototype. Thus, a low-pass m-derived T section can be connected to its constant-k prototype T section without an impedance mismatch occurring at the junction for any frequency.

As has been mentioned before, constant-k sections have poor iterative impedance characteristics, and m-derived 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 mid-shunt iterative impedance of a low-pass m-derived π section is (see page 209 of reference 5)

It is important to note that a, the ratio f/fc enters this equation and to recall that a and m are related as shown in equation 96. Thus, the midshunt iterative impedance of an m-derived low-pass 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 m-derived π section) is substantially constant from zero frequency to cutoff frequency.

It also can be proved that the cutoff frequencies of m-derived sections and their prototypes are identical. (See page 193 of reference 5.)

It will be noted that Figs. 43 and 44 apply both to low-pass and high-pass filters.

The characteristics of constant-k sections, m-derived sections, and their uses in composite filters, will now be summarized. (A) Constant-k sections have poor iterative-impedance characteristics as shown by Figs. 25, 28, and 33. (B) Constant-k 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 m-derived T section is the same, for any value of m, as the midseries iterative impedance of its constant-k prototype; hence, m-derived T sections of any value of m or a and constant-k prototype T sections can be connected together without reflection loss at the junction. (D) The midshunt iterative impedance of an m-derived 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 m-derived and constant-k prototype sections are the same; however, the m-derived section has theoretically infinite attenuation at a frequency determined by a and m.

Based on these facts, composite filters are constructed as follows: Sufficient constant-k 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 m-derived T section is designed, and this is then "split" into two m-derived half sections, which are used to terminate the constant-k sections so that

Figure 44. Variations in the iterative impedance ZKπ for various filter sections with respect to the impedance ZK at f = 0 for the low-pass filter and f = ∞ for the high-pass filter. Frequency ratio f/fc" for low-pass filters, and fc'/f for high-pass filters.

good impedance characteristics and sharp attenuation characteristics are obtained. The m-derived 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 constant-k T sections inside the filter, and hence internal reflections are prevented (Fig. 45).

Figure 45. Illustrating the design of a balanced low-pass composite filter. Inductances in millihenrys. Capacitances in microfarads.

Last Update: 2011-05-30