Introduction To Modern Planar Transmission Lines. Anand K. Verma

      (2.2.9)

      (2.2.10)

      Finally, the voltage distribution on the n^th line section and the voltage at the n^th line junction can be written as follows:

      (2.2.11)

      (2.2.12)

      2.2.2 Location of Sources

      The shunt voltage could be located at any junction and the voltage distribution is computed on any line section due to it. However, it is also interesting to consider a shunt current source and a series voltage source located anywhere on a multisection transmission line. Both kinds of sources create the voltage wave on a line.

      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 I_S 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 Y_L = 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 I_S can be replaced by an equivalent voltage source V_s, shown in Fig (2.11c), at x = 0:

      (2.2.13)

      where Y_in is the total load admittance at the plane containing the current source I_S. 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) = V_s. On using this boundary condition in equation (2.1.79a): V_S = 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 constants V⁺ and V⁻, from these two equations, are obtained as

(2.2.15)

The voltage wave on the left‐hand line #1 is obtained by substituting equation (2.2.15) in equation (2.1.79a):

(2.2.16)

      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⁺ = V_S. 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 V_S 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:

(2.2.18)

      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.

(2 2 19)

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) Скачать книгу