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Lossless Umlonn Transmission Lines 185
This has been programmed efficiently in Program A6-2; given a value of
Z,=R,+jX" the sequence X" R" IJ degrees, and the real number Zo are
entered in tbe stack. Then pressing key A evaluates the input impedance
Z, = R\ + jX\, where R\ is in the X register and XI is in the Y register. The two
cases in Example 6.5 are easily confirmed by this program.
6.2.2. Complex Sources and Complex Loads. Jasik (1961) and Milligan
(1976) have given expressions for finding the Zo and the length of a transmission
line that transform complex Z, to complex input Z\ impedances. Day
(1975) has described a Smith chart method, which will not be considered here.
Moving the denominator of (6.27) to the left side and equating real and
imaginary parts, respectively, yields the desired expressions. The real part is
The imaginary part is
Equation (6.29) yields
where
R\Zo-RlyX,-X\yR,=ZoR,. (6.29)
y(R,R, - XIX, -~) =Zo(X, - X,),
(6.30)
(6.31 )
R\-R,
q= (6.32)
R,X 2 +X,R 2 .
From (6.31) and (6.28), the electrical length of the required transmission line is
IJ= tan -\(Zoq). (6.33)
Substitution of (6.31) for y in (6.30) produces an expression for 20; further
elimination of q, using (6.32) and some algebra, yields the characteristic
impedance of the required transmission line:
Z =(R\IZ212_R,IZd2)'/2
o R,_R,' (6.34)
when R2*R\, and the square root exits.
These relationships have been included in Program A6-2 on key B. The
desired X2, R" X" and R, sequence is entered in the stack. Pressing key B
provides Zo in register X and IJ degrees in register Y if a solution exists.
Otherwise, an error indication is displayed when the HP-67/97 attempts to
compute the square root of a negative number.
Example 6.6. Specifying Z2= IO+j20 and Z\ =30-j40 ohms requires a
matching line with 20=22.36 ohms and IJ=65.91 degrees; this can be checked
using the input impedance calculation on key A. If Z2 is changed to lO+j30
with the same Z\, no match is possible. However, Z2 can first be rotated by 45
degrees on a 50-ohm line; key A shows the resulting impedance to be