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Unifonn T/'Qnsmission Lines 83
for the open- and short-circuited-stub cases, respectively. It is possible to
avoid (4.16) and (4.17) entirely by the identity
. tanh(NP) +jy
tanh(NP+JO)= l+jytanh(NP)' (4.21)
where the definition for y will occur in many places throughout this book:
y=tanO. (4.22)
There is no point in stopping with this exact result for two reasons. First,
short- or open-circuit terminations in the real world are only approximate.
Second, this fact provides an opportunity to drastically shorten the calculations
in hoth programming steps and execution time. Consider the series
approximation
tanhNP=NP- Nt + 1~Np5- .... (4.23)
For dissipation less than I dB, the second term is less than 1/226 of the first
term. An approximate stub input impedance expression is obtained for these
assumptions:
K+jy
Z'n=ZO I +jKy ;
y=tanO;
K= {NP for SC,
NP- l forOC.
(4.24)
Clearly, only one subroutine with a flag variable will suffice for hoth kinds of
stubs in ladder network analysis, using either exact Equations (4.19)-(4.21) or
approximate Equation (4.24) with (4.22). Component menu descriptions with
sample data will be given in Section 4.3.4.
4.3.3. Lossy Transmission Lines in Cascade. Section 4.2.2 shows that parameters
A and D are required for normalization purposes in the L-section
formulation for emhedded ABCD two-port subnetworks. They are equal,
according to (4.13) and (4.17). Furthermore, the L-section branches are
~ =Zotanh(NP+jO),
g=Yotanh(NP+jO),
(4.25)
(4.26)
where the considerations above for the tanh function apply. The approximate
form is recommended only for the more limited hand-held computers. The
rest of the calculation for the cascaded transmission line is accomplished as
for any cascaded two-port subnetwork (see Section 4.2.2). The transmission
line component topology code will be considered next.
4.3.4. Transmission Line Topology Codes. Each cascade and stub transmission
line component will require five items to describe it: type, nepers loss,