Introduction To Modern Planar Transmission Lines. Anand K. Verma

      Example 3.9

Determine the S‐parameters of the series impedance shown in Fig (3.15). Also, compute the attenuation and the phase shift offered by the series impedance.

      Solution

      To compute S₁₁ that is the reflection coefficient of a network under the matched condition, the port‐2 is terminated in Z_0. Thus, Z_in = Z + Z₀ and the reflection coefficient at port‐1 is

(3.1.61)

Figure 3.15 Network of series impedance.

      Likewise, to compute S₂₂, the port‐1 is terminated in Z_0. It gives Z_out = Z + Z₀ at the port‐2. The S₂₂ is

      (3.1.62)

      The total port voltage at the port‐1 is a sum of the forward and reflected voltages:

      To compute S₂₁, i.e. the transmission coefficient from the port‐1 to the port‐2 under the matched termination, at first, the total port voltage at the port‐2 is obtained:

      Therefore, from equations (i) and (ii):

      However, the port voltage V₂ computed from the port current is

      Finally, S₂₁ is obtained from equations :

(3.1.63)

Equations (3.1.61) and (3.1.63) provide the following relation:

      (3.1.64)

      The [S] matrix of the series impedance is

      (3.1.65)

      The attenuation and phase shift of a signal, applied at the input port‐1 of series impedance Z = R + jX, are computed below.

Using S₂₁ from equation (3.1.63), the attenuation offered by the series impedance is

      (3.1.66)

      The lagging phase shift of the signal at the output port‐2, due to the series element, is

      (3.1.67)

      Example 3.10

Determine the S‐parameter of a shunt admittance shown in Fig (3.16). Also, compute the attenuation and the phase shift offered by the shunt admittance.

      Solution

      The shunt admittance is Y = G + jB. To compute S₁₁, the port‐2 is terminated in Z₀ (=1/Y₀) giving Y_in = Y + Y₀. The reflection coefficient of the shunt admittance under matched termination is

      (3.1.68)

      Likewise, to compute S₂₂ of the shunt admittance, the port‐1 is terminated in Z₀:

      (3.1.69)

      Following the previous case of the series impedance, the S₂₁ is computed:

       Fig (3.16) shows V₁ = V₂; therefore,

      (3.1.70)

Figure 3.16 Network of shunt admittance.

      The [S] matrix of the shunt admittance is

      (3.1.71)

      The attenuation of the input signal due to the shunt admittance is

      (3.1.72)

      The lagging phase shift of the signal at the output port‐2, due to the shunt admittance, is

      (3.1.73)

      Example 3.11

Determine the S‐parameters of a transmission line section, shown in Fig (3.17), with an arbitrary characteristic impedance.

      Solution

      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

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