Читать книгу Introduction To Modern Planar Transmission Lines - Anand K. Verma - Страница 77

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.