Introduction To Modern Planar Transmission Lines. Anand K. Verma

elements. Even in such cases, one can find the overall [ABCD] parameter of a complete network. The input impedance, output impedance, Thevenin and Norton equivalent circuits, and power transfer relation, etc. of a complete circuit 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‐ and , enter the ports and the reflected voltage waves at both the ports‐ and , come out of the ports.

      The forward power, i.e. the incident power entering the port‐1, is

(3.1.26)

In equation (3.1.26b), is the RMS voltage of the voltage wave. In general for the two‐port or N‐port network, the forward power entering the i^th port is written as

(3.1.27)

      The incident power variable a_i at the i^th port is defined in a way that the power entering the port is given by the square of the power variable:

(3.1.28)

Figure 3.9 Two‐port network for evaluation of S‐parameter.

Using equations (3.1.27) and (3.1.28), the power variable a_i is written in terms of the forward RMS voltage at the i^th port

      (3.1.29)

      The forward voltage can also be written in term of the power variable as

(3.1.30)

      The power variable a _i is simply a normalized forward voltage wave, incident on the i^th 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 i^th port, in terms of the incident RMS current , is given below:

      (3.1.31)

      The forward port current in terms of the forward power variable is

(3.1.32)

The multiplication of the voltage and current of equations (3.1.30) and (3.1.32), again provides the forward power, . Thus, the definitions of the power variable both as the normalized voltage wave and as the normalized current wave are consistent. However, one must be careful about the presence of the square root of the characteristic impedance in the numerator and denominator for two definitions.

Consider the reflected power wave at the i^th port with characteristic impedance Z_0i. Figure (3.10) shows that the i^th 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 b_i is

Figure 3.10 A section of the multiport network. Port is shown extended with length ℓ = 0.

      (3.1.33)

      The reflected power