Introduction To Modern Planar Transmission Lines. Anand K. Verma

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first isolated line section. The voltage and current waves, with respect to the origin at the load end x₁, on the line section (x₀ ≤ x ≤ x₁) are written from equation (2.1.88):

      (2.2.1)

      The reflection coefficient Γ₁ at the load end, i.e. at x = x₁ is given by

      (2.2.2)

      The load at the x = x₁ end is formed by the cascaded line sections after location x = x₁. The voltage amplitude V⁺ is evaluated by the boundary condition at the input, x = x_0, of the first line section_. At x = x_0, shown in Fig (2.10b), the source voltage is and V⁺ is evaluated as follows:

      (2.2.3)

      The voltage wave on the transmission line section #1 is

(2.2.4)

      The above expression is valid over the range x₀ ≤ x ≤ x₁. The voltage at the output of the line section #1 (x = x₁), that is at the junction of line #1 and line #2, is

(2.2.5)

where d₁ = x₁ − x₀ is the length of the line section #1. The above voltage is input to the line section #2. Equations (2.2.4) and (2.2.5) apply to any line section and at any junction. The voltage is treated as the input voltage of the n^th line section. It is the same as the output voltage of the (n − 1)^th line section. The line length d₁ and reflection coefficient Γ₁ are replaced by d_n and Γ_n, respectively. The cascaded line sections to the right‐hand side of any junction act as a load at the junction and the reflection coefficient at the junction is

      (2.2.6)

      Equation (2.2.4) is applied to Fig (2.10a) to compute the voltage distribution on any line section. The voltage on line section #2 is

      (2.2.7)

The voltage at the output of the line section #2, i.e. the junction voltage of the line sections #2 and # 3 at x = x₂, is obtained from the above equation:

      (2.2.8)

      Using equation (2.2.5) and above equations, the voltage distribution on the line section #2, and also the junction voltage at x = x₂, are obtained:

      (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

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