Introduction To Modern Planar Transmission Lines. Anand K. Verma

port, a part of the incident power appears as the reflected power. The transmission coefficient, S_ij for the power transfer from the j^th port to the i^th port is defined as

(3.1.42)

      Normally, the network has identical port impedances and equal to the system impedance, i.e. Z_0i = Z_0j = Z₀. Equation (3.1.40) is written in compact form as

      (3.1.43)

The elements of the [S] matrix are determined using equations (3.1.41) and (3.1.42).

      Properties of [S] Matrix

      Some important properties of the [S] matrix description of the network are summarized below, without going for the formal proof of these statements. Usually, elements of the [S] matrix are complex quantities. The detailed discussion is available in the well‐known textbooks [B.1–B.5, B.7].

       Reciprocity Property

      The [S] matrix of a reciprocal network is a symmetric matrix, i.e. the transpose [S]^T of the [S] matrix is equal to the [S] matrix itself:

      (3.1.44)

       Unitary Property

      The [S] matrix of a lossless network is a unitary one. However, if the network is not lossless, then it is not unitary. The definition of the unitary matrix provides the following relation for the given [S] matrix:

(3.1.45)

      where [S]^T is the transpose of the [S] matrix, [S]^* is a complex conjugate of the complex [S] matrix and [I] is the identity matrix. Thus, for a given 2‐port [S] matrix, we have

On substituting these expressions in the unitary relation (3.1.45), the following result is obtained:

(3.1.46)

On equating each element of matrix equation (3.1.46), the following relations are obtained:

(3.1.47)

Equations (3.1.47) are generalized for the N‐port network:

(3.1.48)

(3.1.49)

Equation (3.1.48) shows that both elements have identical columns, whereas in equation (3.1.49) column are not identical. The [S] matrix is formed by the column vector as follows:

      (3.1.50)

      Therefore, in the usual vector notation we have

      (3.1.51)

      Hence, for a lossless network the following statements, based on equations (3.1.48) and (3.1.49) are made:

       The dot product of any column vector with its complex conjugate is unity,

       The dot product of any column vector with the complex conjugate of any other column vector is zero,

       The [S] matrix forms an orthogonal set of the vectors.

      The following expressions are written from equation (3.1.47):

(3.1.52)

Equation (3.1.52) is the power balance equations for the lossless two‐port networks. The unit input power fed to the port‐1 is a sum of the reflected power ( |S₁₁|² ) at the port‐1 and the transmitted power |S₂₁|² to the port‐2. In the case |S₁₁|² + |S₂₁|² is less than unity, some power is lost in the network through the mechanism of conductor, dielectric, and radiation losses. The lost power, i.e. the power dissipation in the network, is

      (3.1.53)

       Phase Shift Property

The [S] parameter is a complex quantity. It has both magnitude and phase. Thus, the [S]‐parameter is always defined with respect to a reference plane. In Fig (3.13) [S]‐parameter of the N‐port network is known at the location x = 0. It is determined at the new location, x = −ℓ_n. Alternatively, once the [S] parameters are known at x = −ℓ_n, these are determined at x = 0, i.e. at the port of the network. The location ℓ_n shows the length of the line connected to each port of an N‐port network. Normally, it is the point of measurement of the [S] parameters of the network or device. The interconnecting transmission line is lossless and has propagation constant β_n. Thus, the electrical length of the connecting line is θ_n = β_n ℓ_n.

Figure 3.13 N‐port network showing phase‐shifting property.

      For an N‐port network,