Introduction To Modern Planar Transmission Lines. Anand K. Verma

that supports various kinds of wave phenomena.

      Objectives

       To review the matrix representations of the two‐port networks using the [Z], [Y], and [ABCD] parameters.

       To discuss the basic properties and use of the scattering [S] parameters.

       To understand the process of de‐embedding of true [S] parameters of a device.

       To understand the process of extraction of the propagation constant from the [S] parameters.

       To understand the phase and group velocities in a dispersive medium.

       To discuss the circuit modeling of the reactively loaded line supporting both the forward and backward waves.

3.1 Matrix Description of Microwave Network

      At low frequency, the circuit is described in terms of several kinds of matrices that relate the port voltages to the port currents. These matrices could be the impedance matrix [Z], admittance matrix [Y], and hybrid matrix [H]. The transmission matrix is defined as the [ABCD] matrix. It is useful in cascading of two or more networks or transmission line sections. At low radio frequency, the voltage and current are measurable parameters. Therefore, the matrix elements of a network and device could be experimentally determined.

      Normally, the microwave passive components, circuits, and networks are constructed around the transmission lines supporting the TEM or the quasi‐TEM mode. Sometimes, the lumped elements are also used. The waveguide sections supporting non‐TEM mode are also used to develop components and circuits. As a matter of fact, the voltage and current can be uniquely defined only for the TEM mode supporting structures. However, for non‐TEM line structures, only the equivalent voltage and current, based on the power equivalence principle, is defined [B.1, B.2].

      The abovementioned parameters are discussed in this section, as these are important for the analysis of the line networks and the networks involving both the line sections and lumped circuit elements. The results of the analysis and measurement are also presented using these parameters. The reader can study these parameters in detail from any of the excellent textbooks [B.1, B.3–B.7]. One basic difference could be seen between the lumped elements based low‐frequency circuits and the transmission line sections based on high‐frequency circuits. The low‐frequency circuits are the oscillation type circuits, whereas the high‐frequency microwave circuits are the wave type circuits. In the case of the low‐frequency oscillation type circuits, the port voltage and port current are described by a single voltage or current. In general at any port, for the high‐frequency wave‐type circuits, the port voltage is described by a sum of the incident and reflected voltages, also the port current is a sum of the incident and reflected currents. It is illustrated in the discussions on the evaluation of the parameters.

      How to characterize the components, circuits, and network made of the transmission line sections and waveguide sections? At the microwave frequency, port voltage and port current are not the measurable quantities. However, from the analysis point of view, the networks can be characterized by the [Z], [Y], and [ABCD] parameters. But these are not the measurable parameters at microwave frequencies. A different kind of matrix parameter, called the scattering or [S]‐parameters, is used for the practical characterization of the microwave network and the transmission line structures [J.1]. The [S] parameter is a measurable quantity. A Scalar Network Analyzer is used to measure the magnitude |S| of S‐parameter of any microwave circuit and network. For the measurement of the complex [S] parameters, i.e. both the magnitude and phase response of a network, a Vector Network Analyzer (VNA) is used. The Circuit Simulators and the EM‐Simulators (Electromagnetic field simulators) are also used to get the frequency‐dependent [S] parameters response of the microwave circuits.

      

      3.1.1 [Z] Parameters

      The [Z] matrix defines the impedance parameter of a two‐port or a multiport network. The matrix elements are evaluated by open circuiting the ports. Therefore, the [Z] parameters or the impedance parameters are also called the open‐circuit parameters. The port voltage (V_N) and the port current (I_N) are the sums of the reflected and incident voltages and currents, respectively. The port current is an independent variable, whereas the port voltage is the dependent variable. Therefore, the port currents are the excitation sources creating the port voltages as the response. The response voltage is proportional to the excitation current and the proportionality constant has the dimension of impedance.

The impedance matrix could be obtained for a general linear two‐port network, shown in Fig (3.1). The wave entering the port is the incident voltage () or the incident current () wave. The reflected voltage () and reflected current () waves are also present at the ports. The total port voltage (V_n) or current (I_n) is the sum of the incident and reflected voltage or current:

(3.1.1)

In equation (3.1.1), n = 1, 2 is the port number, i.e. port‐1, port‐2. The power entering the network is taken as a positive quantity for the incident wave, so the power coming out of the network, i.e. the power of the reflected wave, is taken as a negative quantity. The reflected voltage () is positive. To maintain the negative direction of power flow, the reflected current wave () is taken as negative in the above equation. At each port, the current entering the port from outside is positive, whereas the current leaving the port is negative. For the linear networks, the voltage at any port is a combined response of the currents applied to all ports. On using the superposition of the voltage responses, the following set of equations is written:

Figure 3.1 Two‐port network to determine [Z] and [Y] parameters.

(3.1.2)

Equation (3.1.2) is written in a more compact matrix form:

      (3.1.3)

      where [V] and [I] are the column matrices. The two‐port impedance matrix is

      (3.1.4)

      The [Z] parameter can be easily extended to the N‐port networks [B.1, B.3–B.5]. The Z‐parameters are the open‐circuited parameters. The coefficient