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Narrow·Band L, T, and Pi Networks 173
R 1 < R~
Type A
Figure 6.2.
180" ,;;;; phase";;; +180 0 ---,,-__...
Type B (not 0")
Four lumped-element reactance configurations.
the direction of resistance match, as indicated; i.e., a type-A L section can
only decrease the resistance level. Finally, the transfer phase is defined as the
angle by which current I, lags II in Figure 6.1. For resistive Z, and Z" this is
the same as the phase lag of V, with respect to VI' The phase angles of type-A
and type-B L sections are dependent, assuming that the R I and R 2
terminations
are independent. However, the phase is also independent for the T and
pi sections shown in Figure 6.2 within the range indicated.
It has been remarked that the phase is not of interest in many cases;
however, it is a convenient parameter and represents a degree of freedom for
T and pi networks. (The phase sign is a degree of freedom for L sections.) It is
also noted that L, T, and pi sections may be designed by the I +Q' method
described in Section 6.1.3 without consideration of phase. The reader is
expected to adopt the techniques most useful for his purpose.
This topic will be developed by first considering the interface impedances
resulting from the use of a lossless network, especially the relationship between
Zq and Z, and between Z, and Z, in Figure 6.1. Then the basic case for T and
pi network matching from resistance R, and R, will be given. The L sections
will be special cases of these, in which branch X 3
in Figure 6.2 is removed. To
accommodate complex source and load impedances, series-to-parallel impedance
conversions and the opposite case will be developed. Also, the impedance
of paralleled impedances will be discussed. These conversions adapt complex
terminations to the prior analysis for resistive transformations. Finally, the
role of graphic procedures-especially the Smith chart-will be considered in
some detail. Programs are provided.
6.1.1. Lossless Network Interface Impedances. There is an important impedance
concept associated with maximum power transfer by a lossless network
of any type. Consider the power transferred from the source to the network in
Figure 6.1. According to the analysis in Section 3.2.3, maximum available
power (P a ,) is transferred when ZI =Z:. For lossless networks, the maximum
available power must arrive undiminished at the load end of the network,
where the Thevenin impedance looking back into the network is Zq. At the
load port, then, there must be an equivalent Thevenin source providing the
same maximum available power; therefore, Z, = Z~. In fact, the matched,
lossiess network can be cut at any interface and a conjugate match will exist;