Anand K. Verma - Introduction To Modern Planar Transmission Lines

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.

To compute S 11that is the reflection coefficient of a network under the matched condition, the port‐2 is terminated in Z 0.Thus, Z in= Z + Z 0and the reflection coefficient at port‐1 is

(3.1.61)

Figure 3.15 Network of series impedance.

Likewise, to compute S 22, the port‐1 is terminated in Z 0.It gives Z out= Z + Z 0at the port‐2. The S 22is

(3.1.62)

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

To compute S 21, 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 V 2computed from the port current is

Finally, S 21is 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 S 21from 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)

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.

The shunt admittance is Y = G + jB. To compute S 11, the port‐2 is terminated in Z 0(=1/Y 0) giving Y in= Y + Y 0. The reflection coefficient of the shunt admittance under matched termination is

(3.1.68)

Likewise, to compute S 22of the shunt admittance, the port‐1 is terminated in Z 0:

(3.1.69)

Following the previous case of the series impedance, the S 21is computed:

Fig (3.16)shows V 1= V 2; 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)

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

The line has an arbitrary characteristic impedance nZ 0and propagation constant β. The Z 0is 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 nZ 0is

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

(3.1.75)

Thus, the reflection coefficient at the port‐1 is