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174 Impedance Matching
i.e., the pertinent generalized reflection coefficient magnitude (3.46) must be
zero at every interface. If there is a mismatch anywhere, then the pertinent
reflection coefficients at every interface must all have the same magnitude,
since the actual and available power are the same everywhere.
The most elementary application occurs when only resistances are anticipated
at th~ lossless network terminals in Figure 6.1. Then Zq=R,=Z2 implies
that Z,= R, =Z,. Especially, it is common to talk about resistive terminations
where the simple fact that Z, = R, would not necessarily imply that Z,= R,.
This small nuance arises in Section 6.1.4 when matching from a complex load
to a possibly complex source.
6.1.2. Real Source and Real Load. The reactance equations for T and pi
networks are given in Table 6.1, along with the modifications for the L
sections. T-section matching relationships will be verified; the pi-section
relationships can be verified similarly.
Consider Figure 6.3. Define
~=~~~. (~~
Replacing X, in (6.9) with its T-section expression (6.5) from Table 6.1 yields
If 12= 1+jO in Figure 6.3, then V' = Zo and
~R,R2
R2e jP -
Zo = j sin f3 (6.10)
Zo
I, =.,.-X + I. (6.11)
J 2
Table 6.1.
T, Pi, and L Reactance Equations
T
Pi
X,= VR,R2 - R, cos f3 sin f3
(6.1 ) X j =R\R
sin {1
2
R2cos f3-VR,R2
(6.2)
VR,R2 sin ,8
X2=---- (6.3) X2=R,R2--- (6.4)
sin p
VR,R2
X,= VR,R 2 - R2cos f3
sin f3
sin f3
(6.5) X,=R,R2
(6.6)
R, cos f3 - VR,R2
. r;:: ~
For LA: use T wIth X 3 =O; {3= ± cos- l VR;"' =tan- I V1f;-1
For LB: use pi with X 3 -,)ooo; f3 as above, with R 1 and R 2
exchanged.
(6.7)
(6.8)