interconnecting transmission line is lossless and has propagation constant βn. Thus, the electrical length of the connecting line is θn = βn ℓn.
Figure 3.13 N‐port network showing phase‐shifting property.
For an N‐port network, the incident wave at the nth port, x = − ℓn, after reflection from the port at x = 0, returns to x = − ℓn. In the process, it travels the electrical length 2θn. Similarly, if the wave is incident at port #1, located at x = − ℓ1 and arrives at the port‐2, located at x = − ℓ2; the electrical length traveled by the wave is θ1 + θ2 = β1 ℓ1 + β2 ℓ2, or 2θ1, on the assumption that β1 = β2, and ℓ1 = ℓ2, i.e. the transmission lines connected at both the ports are identical. The measured or simulated scattering matrix [S′] at the location x = − ℓn is related to the [S] parameters of the network by the following expression
The [S]‐parameter of the network is extracted from equation (3.1.54), as
(3.1.55)
For reducing the cascaded network to a single equivalent network, the [S] parameters cannot be cascaded like the [ABCD] parameters. The [ABCD] matrix is suitable for this purpose. However, it is not defined in terms of the power variables. Therefore, another suitable transmission matrix, called [T] matrix has been defined in terms of the power variables to cascade the microwave networks. The [S] matrix is easily converted to the [T] parameters [B.1, B.2–B.5, B.7, B.9].
The concept of the [S] matrix is used below to some simple, but useful circuits. These examples would help to appreciate the applications of the [S] parameters.
Example 3.8
Determine the S‐parameters and return loss of a 2‐port network with arbitrary termination shown in Fig (3.14).
Solution
The 2‐port network (device) is connected to a source at the port‐1 and a load ZL at the port‐2. The source has voltage Vg with internal impedance Zg. The network scattering parameters‐[S] are computed under the matched condition. The characteristic impedance of the connecting line between the port‐1 and the source is Z01, whereas the characteristic impedance of the connecting line between the port‐2 and the load is Z02. The lengths of the connecting lines are zero. The reflection and transmission coefficients are to be determined at the input and output terminals. This is a practical problem for the measurement and simulation of the 2‐port network:
Figure 3.14 A two‐port network with arbitrary termination.
Figure (3.14) shows that the power variable b2 is the incident wave at the load ZL and the power variable a2 is the reflected wave from the load. Thus, the reflection coefficient at the load is
From above equations (i) and (ii):
(iii)
On substituting b2 from equation (b) in equation (a):
(iv)
The input reflection coefficient at the port‐1 is
(3.1.56)
The reflection coefficient Γ1 is more than S11 of the network. The mismatch at the load degrades the return loss (RL) of the network. It is given by
(3.1.57)
For the port 2 open‐circuited (ZL → ∞), the waves get reflected in‐phase, i.e. ΓL = 1, and for a short‐circuited load (ZL = 0) the total reflection is out of phase, i.e. ΓL = −1. If the network is terminated in a matched load (ZL = Z02), the incident waves are absorbed with ΓL = 0 and Γ1 = S11. Likewise, the source reflection coefficient Γg could be defined at the input port‐1. Figure (3.14) again shows that b1 is the incident wave on the internal impedance of the source Zg and a1 is the reflected wave from Zg. Thus,
The output reflection coefficient Γ2 at the port‐2 is obtained from equations (i) and (v):
(3.1.58)
Again under the matched condition (Zg = Z01) at the input port, Γg = 0. For most of the applications, 50 Ω system impedance is used, i.e. Z01 = Z02 = Z0 = 50 Ω. For a 2‐port lossless network, we have the following expressions:
However, for a reciprocal network S12 = S21. Thus, the above equations provide
(3.1.59)
