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
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
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
The port voltage V1 is obtained as a sum of the incident and reflected voltages at the port‐1:
At the port‐2, under the matched termination, ZL = nZ0 giving
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.
3.2 Conversion and Extraction of Parameters
Sometimes, the conversion of one kind of network parameter to another kind is needed for the analysis of a circuit. For instance, if several circuit blocks comprising of the lumped elements and the transmission line sections are cascaded, each circuit block could be expressed by its [ABCD] matrix. It helps to get an overall [ABCD] matrix of the cascaded network. However, the final [ABCD] matrix, describing the cascaded network is further converted to the [S] matrix. Similarly, the [S] matrix of each building block of the cascaded network has to be converted to the [ABCD] matrix to get the overall [ABCD] matrix of the cascaded network. Finally, the overall [ABCD] matrix is converted to the [S] matrix of the cascaded network. The S‐parameters are measurable quantities. The performance of a network is measured in the [S] matrix using a VNA.
On several occasions, the S‐parameters of a line section or a network are known either from the simulations or from the measurements. The S‐parameters are used to get the characteristic impedance and the propagation constant of a line, or a network. However, the true S‐parameters of a network are needed for this purpose. The true S‐parameters are normally embedded in the measured or the simulated S‐parameters at the ports of measurement, or the ports of simulation. The true S‐parameters of a line or a network are extracted, i.e. de‐embedded, from the measured, or simulated, S‐parameters at the ports. This is known as the de‐embedding process [B.10]. The EM‐Simulators have provision to de‐embed the true S‐parameters from the S‐parameters obtained at the measurement or simulation ports.
This section presents the conversion of matrix parameters, de‐embedding of the S‐parameters, and extraction of the propagation characteristics.
3.2.1 Relation Between Matrix Parameters
[Z] and [ABCD] Parameters
