Anand K. Verma - Introduction To Modern Planar Transmission Lines

The voltage and current on a transmission line can be written as

The voltage and current at the input port‐1 are obtained for x = −ℓ:

Above equations can be written in the matrix form:

The [ABCD] parameters of the lossy and lossless transmission line sections are given by equation (3.1.25a)and equation (3.1.25b),respectively:

(3.1.25)

The above example can be further extended to a network of several cascaded transmission line sections having different ℓ, Z 0, and γ. The overall [ABCD] parameter of the multisection transmission line can be obtained by a multiplication of the [ABCD] matrix of each line section. The line sections can be attached to the series and the shunt lumped 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].

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].

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 Z 01and Z 02, respectively. However, most of the two‐port networks have Z 01= Z 02= Z 0, i.e. the identical transmission line sections at both the ports. The reference impedance Z 0is 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 thport is written as

(3.1.27)

The incident power variable a iat the i thport 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 iis written in terms of the forward RMS voltage at the i thport

(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 thport. 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 thport, in terms of the incident RMS current , is given below:

(3.1.31)