文档介绍：IMPEDANCE TRANSFORMATION EQUATION 101
IMPEDANCE TRANSFORMATION EQUATION
One of the mon tasks in microwave engineering is the determination
of how a load impedance ZL is transformed to a new input impedance ZIN by a
length of uniform transmission line of characteristic impedance Z0 and electri-
cal length y (Fig. -1).
To simplify this derivation, we assume that the line length is lossless. With
the choice of x ¼ 0 at the load, the input to the line is at x ¼Àl, and the input
impedance there is
V ðx ¼ÀlÞ e jbl þ G eÀjbl
Z Z x l Z L 4:14-1
IN ¼ ð ¼ÀÞ¼ ¼ 0 jbl Àjbl ð Þ
I ðx ¼ÀlÞ e À GLe
jbl
Now substitute GL ¼ðZL À Z0Þ=ðZL þ Z0Þ, bl ¼ y, the identities e ¼
cos bl þ j sin bl and eÀjbl ¼ cos bl À j sin bl into (-1), and remove cancel-
ing terms to get
2ZL cos y þ j2Z0 sin y
ZIN ¼ Z0
2Z0 cos y þ j2ZL sin y
ð4:14-2Þ
ZL þ jZ0 tan y
ZIN ¼ Z0
Z0 þ jZL tan y
Similar reasoning can be used to evaluate the input impedance when the trans-
mission line has finite losses. The result is
ZL þ Z0 tanh gl
ZIN ¼ Z0 ð4:14-3Þ
Z0 þ ZL tanh gl
This is one of the most important equations in microwave engineering and
is called the impedance transformation equation. Remarkably, (-2) and
(-3) have exactly the same format when derived in terms of admittance. For
the lossless line
YL þ jY0 tan y
YIN ¼ Y0 ð4:14-4Þ
Y0 þ jYL tan y
Figure -1 Equivalent circuit of lossless uniform transmission line of electrical
length y and characteristic impedance Z0 terminated in impedance ZL.
102 DISTRIBUTED CIRCUITS
and for the line with loss
YL þ Y0 tanh gl
YIN ¼ Y0 ð4:14-5Þ
Y0 þ YL tanh gl
where Y0 ¼ 1=Z0, YL ¼ 1=ZL, and YIN ¼ 1=ZIN. Expressions (-4) and
(-5) can be verified by substituting these equivalences, respectively, into
(-2) and (-3).
In general, even the lossless expressions of (-2) and (-4) plex
to apply, as wil