Page 354 - Engineering Electromagnetics, 8th Edition

336 ENGINEERING ELECTROMAGNETICS

In polar form, we have used | | and φ as the magnitude and angle of .With r and
i as the real and imaginary parts of ,we write
(108)
= r + j i
Thus
1 + r + j i
r + jx = (109)
1 − r − j i
The real and imaginary parts of this equation are

2
1 − − 2 i
r
r = (110)
2
(1 − r ) + 2 i
2 i
x = 2 (111)
2
(1 − r ) + i
After several lines of elementary algebra, we may write (110) and (111) in forms
which readily display the nature of the curves on r , i axes,
r 1
2 2
2
r − + = (112)
i
1 + r 1 + r
1 1
2 2
2 (113)
( r − 1) + i − =
x x
The first equation describes a family of circles, where each circle is associated
with a specific value of resistance r.For example, if r = 0, the radius of this zero-
resistance circle is seen to be unity, and it is centered at the origin ( r = 0, i = 0).
This checks, for a pure reactance termination leads to a reflection coefficient of unity
magnitude. On the other hand, if r =∞, then z L =∞ and we have = 1 + j0.
The circle described by (112) is centered at r = 1, i = 0 and has zero radius. It is
therefore the point = 1 + j0, as we decided it should be. As another example, the
circle for r = 1is centered at r = 0.5, i = 0 and has a radius of 0.5. This circle
is shown in Figure 10.10, along with circles for r = 0.5 and r = 2. All circles are
centered on the r axis and pass through the point = 1 + j0.
Equation (113) also represents a family of circles, but each of these circles is
defined by a particular value of x, rather than r.If x =∞, then z L =∞, and
= 1 + j0again. The circle described by (113) is centered at = 1 + j0 and has
zero radius; it is therefore the point = 1+ j0. If x =+1, then the circle is centered
at = 1 + j1 and has unit radius. Only one-quarter of this circle lies within the
boundary curve | |= 1, as shown in Figure 10.11. A similar quarter-circle appears
below the r axis for x =−1. The portions of other circles for x = 0.5, −0.5, 2,
and −2 are also shown. The “circle” representing x = 0is the r axis; this is also
labeled in Figure 10.11.
The two families of circles both appear on the Smith chart, as shown in
Figure 10.12. It is now evident that if we are given Z L ,we may divide by Z 0 to