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 (VN) and the port current (IN) 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 (
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 (
Figure 3.1 Two‐port network to determine [Z] and [Y] parameters.
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 of the matrix can be defined in terms of the open circuit condition at the ports:
(3.1.5)
All the ports are open‐circuited, except the ith port at which the matrix element Zii is defined. For instance, in the case of a two‐port network, Z11 is obtained when current I1 is applied to port‐1 and the voltage response is also obtained at the port‐1, while keeping the port‐2 open‐circuited, i.e. I2 = 0. The coefficient, Z11, is known as the self‐impedance of the network. These are the diagonal elements of a [Z] matrix. The off‐diagonal elements of a [Z] matrix are defined as follows:
(3.1.6)
In this case, the current excitation is applied at the port‐j and the voltage response is obtained at the port‐i. All other ports are kept open‐circuited allowing Ik = 0, except at the port‐j. For instance, in the case of a two‐port network to evaluate Z12, the current source is applied at the port‐2, and the voltage response is obtained at the port‐1, while keeping the port‐1 open‐circuited. The coefficient Z12 is the mutual impedance that describes the coupling of port‐2 with the port‐1. A network can have Z11 = Z22, i.e. both of the ports are electrically identical. Such a network is known as the symmetrical network.
