alt="equation"/>
The network also follows
(3.1.60)
Therefore, once the complex S11 and S22 are measured, both the magnitude and phase of the S21 are determined. However, usually, both S11 and S21 are obtained from a VNA and also from the circuit simulator or EM‐simulator. The magnitude of S21 provides the insertion‐loss of the network and ϕ is the phase shift at the output of the network.
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 Z0. Thus, Zin = Z + Z0 and the reflection coefficient at port‐1 is
Figure 3.15 Network of series impedance.
Likewise, to compute S22, the port‐1 is terminated in Z0. It gives Zout = 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 :
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 Yin = 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:
(3.1.70)
Figure 3.16 Network of shunt admittance.
The [S] matrix of the shunt admittance is
(3.1.71)
