can be written in terms of the [ABCD] matrix. However, a detailed discussion of these aspects is out of the scope of this book. The reader can follow many available texts for this purpose [B.1, B.2–B.5, B.7, B.8].
3.1.4 Scattering [S] Parameters
The [Z], [Y], and [ABCD] matrix descriptions of any microwave network or component are based on the port voltage and port current relations. The evaluation of these parameters requires the short‐circuiting and open‐circuiting of the ports. At the microwave frequency, usually, it is difficult to measure the voltage and current. Similarly, the short‐circuiting and open‐circuiting of the ports may not be always possible at the microwave frequency. Thus, these parameters are normally not measurable quantities. However, these parameters are useful for the analysis of the microwave circuits built around the lumped and distributed circuit elements. At this stage, another kind of measurable parameters is needed to characterize the microwave circuits. At the microwave frequency, the power could be measured, and also the forward and reflected power waves could be obtained. The frequency and phase of a microwave signal are also measurable quantities. The scattering parameters, also called the S‐parameters, are defined for any two‐port network, or even the multiport microwave network, in terms of the measurable incident and reflected power waves [J.1].
Basic Concept
A commonly used two‐port network is suitable to develop the concept of the S‐parameter. Even the S‐parameters of a multiport network are measured as the two‐port parameters, while other ports are terminated in the matched loads. Figure (3.9) shows the two‐port network. It is to be characterized by the S‐parameters. The port‐1 and port‐2 are terminated with the line sections of characteristic impedance Z01 and Z02, respectively. However, most of the two‐port networks have Z01 = Z02 = Z0, i.e. the identical transmission line sections at both the ports. The reference impedance Z0 is normally 50Ω. The incident voltage waves at both the ports‐
The forward power, i.e. the incident power entering the port‐1, is
In equation (3.1.26b),
The incident power variable ai at the ith port is defined in a way that the power entering the port is given by the square of the power variable:
Figure 3.9 Two‐port network for evaluation of S‐parameter.
Using equations (3.1.27) and (3.1.28), the power variable ai is written in terms of the forward RMS voltage
(3.1.29)
The forward voltage can also be written in term of the power variable as
The power variable a i is simply a normalized forward voltage wave, incident on the ith port. The normalization is done with respect to the square root of the characteristic impedance at the port. The forward power variable can also be viewed as the incident normalized current. The power entering the ith port, in terms of the incident RMS current
(3.1.31)
The forward port current in terms of the forward power variable is
The multiplication of the voltage and current of equations (3.1.30) and (3.1.32), again provides the forward power,
Consider the reflected power wave at the ith port with characteristic impedance Z0i. Figure (3.10) shows that the ith port is connected to a source with impedance Z0. For the sake of clarity, the port is taken out of the network using an interconnect line of characteristic impedance Z0 with zero length, ℓ = 0. The total power available from the source does not enter the network. A part of it gets reflected. The reflected power in terms of the reflected power variable bi is
Figure 3.10 A section of the multiport network. Port is shown extended with length ℓ = 0.
(3.1.33)
The
