Introduction To Modern Planar Transmission Lines. Anand K. Verma
(2.2.9)
(2.2.10)
Finally, the voltage distribution on the nth line section and the voltage at the nth line junction can be written as follows:
(2.2.11)
(2.2.12)
2.2.2 Location of Sources
The shunt voltage
Current Source at the Junction of Finite Length Line and Infinite Length Line
Figure (2.11a) shows a transmission line circuit with a current source IS located at x = 0 that is the junction of two lines of different electrical characteristics. The open‐circuited line #1, with length x = −d1, is located at the left‐hand side of the current source. Its characteristics impedance/admittance is (Z01/Y01) and its propagation constant is β1. The infinite length line #2, with characteristics impedance/admittance (Z02/Y02) and the propagation constant β2, is located at the right‐hand side of the current source. It can be replaced by a load admittance YL = Y02 at a distance x = d2, shown in Fig (2.11b). The objective is to find out the voltage waves on both the lines as excited by the current source.
Figure 2.11 A shunt current source at the junction of two‐line sections.
The current source IS can be replaced by an equivalent voltage source Vs, shown in Fig (2.11c), at x = 0:
(2.2.13)
where Yin is the total load admittance at the plane containing the current source IS. Y− and Y+ are left‐hand and right‐hand side admittances at x = 0 given by
(2.2.14)
The general solution of a voltage wave is given by equation (2.1.79a). The constants V+ and V− are evaluated for the left‐hand side of a lossless transmission line. At x = 0, V(x = 0) = Vs. On using this boundary condition in equation (2.1.79a): VS = V+ + V−. At x = −d1 the line is open‐circuited with I (x = −d1) = 0. On using this boundary condition in equation (2.1.79b):
The voltage wave on the left‐hand line #1 is obtained by substituting equation (2.2.15) in equation (2.1.79a):
The line at the right‐hand side of the current source is an infinite length line that supports a traveling wave without any reflection. Therefore, at x = 0, V− = 0 and V+ = VS. The voltage wave on line #2 at the right‐hand side is
(2.2.17)
The method can be easily extended to a multisection line structure. For this purpose, the left‐hand and right‐hand side admittances Y− and Y+ are determined at the plane containing the current source.
Series Voltage Source
Figure (2.12a) shows the series‐connected voltage source VS at x = 0. The location x = 0 is a junction of two transmission lines – line #1 open‐circuited finite‐length line and line #2 infinite length line. The lines at the left‐hand and right‐hand sides of the voltage source can be replaced by the equivalent impedances Z− and Z+, respectively. It is shown in the equivalent circuit, Fig (2.12b). Again, the voltage waves on both lines, excited by a series voltage source, could be determined.
The voltages across loads Z− (Z1) and Z+ (Z2), shown in Fig (2.12b), are obtained as follows:
Line #1 is open‐circuited and line #2 is of infinite extent. Therefore, their input impedances at x = 0− and x = 0+ are
Figure 2.12 A series voltage source at the junction of two‐line sections.
The voltage at x = 0+ from equations (2.2.18c) and (2.2.19) is
(2.2.20)
For a lossy transmission line, the above equation could be written as follows:
(2.1.21)