Introduction To Modern Planar Transmission Lines. Anand K. Verma

Скачать книгу

(3.18) shows a network with its known [Z] parameters. It requires conversion to the [ABCD] parameters. The [ABCD] and [Z] parameters of the network are summarized below:

(3.2.1)

(3.2.2)

Figure 3.18 Network for Z‐parameter.

The current (I₂) entering the port‐2 is taken positively. However, the Z‐parameter is defined in Fig (3.18) for the output current leaving the port. In this case, I₂ is negative. Equation (3.2.2) is rearranged to get the port voltage and current V₁ and I₁ at the port‐1 in terms of the V₂ and I₂ at the port‐2:

(3.2.3)

On comparing equations (3.2.1) and (3.2.3), the following conversion expressions are obtained:

      (3.2.4)

      Likewise, the relations between [Y] and [ABCD] parameters are obtained:

      (3.2.5)

      A complete set of the conversion table of parameters is available in textbooks [B.1, B.5, B.7].

      [S] and [Z] Parameters

      The N‐port network, having normalized reference port impedance, Z_0n = 1, is considered. The port voltage and port current in terms of the incident and reflected voltage can be written as

      (3.2.6)

      The above equations are written in the column matrix form:

      (3.2.7)

      The port voltage is related to the port current through the [Z]‐matrix:

      (3.2.8)

      where [I] is a unit or identity matrix. Keeping in view the definition of the [S] matrix, the following relations, between the [S] matrix and [Z] matrix, are obtained:

      (3.2.9)

      Similarly, the following expressions, relating [S] and [Y]‐parameters are obtained:

      (3.2.10)

      [ABCD] and [S] Parameters

Figure (3.19) shows a 2‐port network. The known [ABCD] parameters of the network are to be converted to the [S] parameters. The voltage pair , and are the incident voltage and reflected voltage at both port‐1 and port‐2. Figure (3.19) also shows the total port voltage (V₁, V₂) and total port current (I₁, I₂). The port currents, at both the ports, enter into the network. Therefore, the current I₂ is negative. The input port voltage and port current are related to the output port voltage and port current through the [ABCD] parameters as follows:

(3.2.11)

      The port voltage and current are a linear combination of the incident and reflected voltages and currents:

(3.2.12)

Figure 3.19 Network for [ABCD] parameter.

On substituting equation (3.2.11) in equation (3.2.12):

(3.2.13)

      To define the [S] parameters, port‐2 is terminated in the reference impedance Z₀ giving , i.e. the reflection from the matched terminated load is zero. The voltage is the incident wave on the load (Z_L = Z₀), whereas the voltage is the reflected wave from the load. The above equations are reduced to the following expressions:

(3.2.14)

      On adding the above equations, the following expression is obtained:

(3.2.15)

Equation (3.2.15) provides the transmission coefficient S₂₁, defined as follows:

      (3.2.16)

Скачать книгу