Читать книгу Introduction To Modern Planar Transmission Lines - Anand K. Verma - Страница 111

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)