3-16
The impedance presented to the input terminals of the transmission line is not merely the resistance
of the wire in series with the impedance of the load. The effects of series inductance and shunt
capacitance of the line itself may overshadow the resistance, and even the load, as far as the input
terminals are concerned.
To find the input impedance of a transmission line, determine the impedance of a single section of
line. The impedance between points K and L, in view B of figure 3-16, can be calculated by the use of
series-parallel impedance formulas, provided the impedance across points M and N is known. But since
this section is merely one small part of a longer line, another similar section is connected to points M and
N. Again, the impedance across points K and L of the two sections can be calculated, provided the
impedance of the third section is known. This process of adding one section to another can be repeated
endlessly. The addition of each section produces an impedance across points K and L of a new and lower
value. However, after many sections have been added, each successive added section has less and less
effect on the impedance across points K and L. If sections are added to the line endlessly, the line is
infinitely long, and a certain finite value of impedance across points K and L is finally reached.
In this discussion of transmission lines, the effect of conductance (G) is minor compared to that of
inductance (L) and capacitance (C), and is frequently neglected. In figure 3-16, view C, G is omitted and
the inductance and resistance of each line can be considered as one line.
Let us assume that the sections of view C continue to the right with an infinite number of sections.
When an infinite number of sections extends to the right, the impedance appearing across K and L is Z0.
If the line is cut at R and S, an infinite number of sections still extends to the right since the line is endless
in that direction. Therefore, the impedance now appearing across points R and S is also Z0, as illustrated
in view D. You can see that if only the first three sections are taken and a load impedance of Z0 is
connected across points R and S, the impedance across the input terminals K and L is still Z0. The line
continues to act as an infinite line. This is illustrated in view E.
Figure 3-17, view A, illustrates how the characteristic impedance of an infinite line can be
calculated. Resistors are added in series parallel across terminals K and L in eight steps, and the resultant
impedances are noted. In step 1 the impedance is infinite; in step 2 the impedance is 110 ohms. In step 3
the impedance becomes 62.1 ohms, a change of 47.9 ohms. In step 4 the impedance is 48.5 ohms, a
change of only 13.6 ohms. The resultant changes in impedance from each additional increment become
progressively smaller. Eventually, practically no change in impedance results from further additions to the
line. The total impedance of the line at this point is said to be at its characteristic impedance; which, in
this case, is 37 ohms. This means that an infinite line constructed as indicated in step 8 could be
effectively replaced by a 37-ohm resistor. View B shows a 37-ohm resistor placed in the line at various
points to replace the infinite line of step 8 in view A. There is no change in total impedance.
