Introduction To Modern Planar Transmission Lines. Anand K. Verma

href="#ulink_a16b8d57-a419-549f-a173-f661e4effea1">(2.1.121)

      (2.1.122)

      For a lossless line, the power balance is written as follows:

(2.1.123)

In equation (2.1.123), input power P_in at the input port – aa of the line enters into the line. It is supplied by a source. The output power P_out is the power supplied to the load.

      

2.2 Multisection Transmission Lines and Source Excitation

      This section extends the solution of the voltage wave equation to the multisection transmission line [B.8, B.16]. Next, the voltage responses are obtained for the shunt connected current source, and also the series‐connected voltage source, at any location on a line. This treatment is used in chapters 14 and 16 for the spectral domain analysis of the multilayer planar transmission lines.

      2.2.1 Multisection Transmission Lines

Figure (2.10a) shows a multisection transmission line, consisting of the N number of line sections. Each line section has different lengths (d₁, d₂,…,d_N), different characteristic impedances (Z₀₁, Z₀₂,…,Z_0N), or different characteristic admittances (Y₀₁, Y₀₂,…,Y_0N) and different propagation constants (β₁, β₂, …, β_N). At each junction of two dissimilar lines, the voltage wave reflection occurs with the reflection coefficient Γ₁, Γ₂, …, Γ_N. At each junction, the input admittance of all succeeding line sections appears as a load. The input admittances at junctions (x₁, x₂, …, x_N) are . The distances of the junctions are measured from the origin that is located on the left‐hand side. The input terminals of line sections 1,2,…,N − 1 are located at the junctions (x_0, x₁, x_2,…,x_N). The voltage source V_s is located at the input of the first line section. The last line section is terminated in the load Z_L = 1/Y_L.

Figure 2.10 The multisection transmission line.

      The objective is to find the voltage at each junction of the multisection line. Further, the voltage distribution on each line section is determined due to the input voltage .

      The solutions for the voltage and current wave equations involve four constants. The constants of the current wave are related to two constants of the voltage wave through the characteristic impedance of a line. Out of two constants of the voltage wave, one is expressed in terms of the reflection coefficient at the load end; that itself is expressed by the characteristic impedance and the terminated load impedance. The reflection coefficient can also be expressed by the characteristic admittance and the terminated load admittance. The second constant is evaluated by the source condition at the input end. Figure (2.10b) shows the 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