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94 udder Network Analysis
identity in (4.43) results after some algebra, using the identities
IZI'=ZZ', 2ReZ=Z+Z·. (4.52)
The reflection coefficients looking into port p are defined by
b p
Pp=-'
a p
(4.53)
But definitions (4.48) and (4.49) in (4.53) yield
Zp-Z~*
Pp= Z +Z" .
p p
(4.54)
It is important to note that this is essentially (3.46), which enabled a simple
calculation of power transfer from a complex source to a complex load.
Kurokawa (1965) discusses the differences in reflection of power waves and
traveling waves on transmission lines with real or complex Zo° In general,
traveling waves are not closely related to power.
Finally, the transducer function, S'I' is defined by (4.47):
, S'I=a
l
b'l
a2=O
(4.55)
The side condition that there be no reflection from the load is important in
itself; it requires that ZL =Z,. (Why?) Using (4.48) and (4.49) in (4.55) and
equating a, from (4.48) to zero yields the general transducer function
I+Z,' ~1 V,
821 = zn Rn E; Zs=Z7·
, "
(4.56)
4,5.3. Wave Response Functions. Scattering parameters normalized to complex
port impedances will be used throughout Chapter Seven; the more
familiar case of real port-normalizing impedances will be assumed. Also, the
source impedance will be assumed to be equal to the port-l normalizing
resistance R I . Then, the input reflection coefficient from (4.54) is
ZI-RI
PI=Z +R'
I I
(4.57)
which is the same as (3.48). When ZL = Z, = R,+jO, (4.57) is equal to coefficient
SII in (4.46). The reflection coefficient looking into port 2 is defined in a
similar way, and will be used in Section 6.7. A low reflection coefficient
magnitude indicates a high-quality impedance match as ZI approaches R I .
Three ways to express this co~dition are return loss, standing-wave ratio
(SWR), and mismatch loss.
Return loss is commonly used in microwave design; it is defined to be:
RL= -2010g lO lpl dB. (4.58)