Читать книгу Introduction To Modern Planar Transmission Lines - Anand K. Verma - Страница 120

The line has an arbitrary characteristic impedance nZ₀ and propagation constant β. The Z₀ is taken as the reference impedance to define the S‐parameter. The reflection coefficient at the load end is

(3.1.74)

Using equation (2.1.88) of chapter 2, the input impedance at the port‐1 of the transmission line having characteristic impedance nZ₀ is

Figure 3.17 A transmission line circuit with an arbitrary characteristic impedance.

(3.1.75)

Thus, the reflection coefficient at the port‐1 is

(3.1.76)

The transmission parameter S₂₁ is computed in terms of S₁₁. If the amplitude of the forward traveling voltage wave on the transmission line is , the total voltage on the transmission line is given by

(3.1.77)

where x is measured from the load end, as shown in Fig (3.17). The input port‐1 is located at x = − ℓ. The voltage at the port‐1 is

(3.1.78)

The port voltage V₁ is obtained as a sum of the incident and reflected voltages at the port‐1:

(3.1.79)

At the port‐2, under the matched termination, Z_L = nZ₀ giving and . Equation (3.1.77) shows that the voltage at the port‐2, i.e. at x = 0 is

(3.1.80)

Using equation (3.1.79), the transmission coefficient, S₂₁ of the circuit shown in Fig (3.17) is obtained as

(3.1.81)

On substituting V₁ from equation (3.1.78) and V₂ from equation (3.1.80) in the above equation S₂₁ is obtained:

(3.1.82)

The present line network is symmetrical and reciprocal. It has S₁₁ = S₂₂ and S₂₁ = S₁₂. The above expressions are checked for n = 1, i.e. for a transmission line of characteristic impedance Z₀. For this case, S₁₁ = S₂₂ = 0 and S₂₁ = S₁₂ = e^{−j βℓ}. These are expressions of the S‐parameters for a line having characteristic impedance Z₀.