Introduction To Modern Planar Transmission Lines. Anand K. Verma

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)

The following expression is obtained from equation (3.2.14):

(3.2.17)

Equation (3.2.17) gives the following reflection coefficient S₁₁ at the port‐1, while port‐2 is terminated in the matched load Z₀:

      (3.2.18)

Similarly, the following expressions are obtained from equation (3.2.13), where the port‐1 is terminated in the matched load Z₀, i.e. :

      (3.2.19)

      On eliminating the S₂₂ is obtained, whereas S₁₂ is obtained eliminating :

      (3.2.20)

      If the network is reciprocal, AD − BC = 1, i.e. S₁₂ = S₂₁. For the symmetrical network, S₁₁ = S₂₂ leading to A = D. The known [S] parameters can also be converted to the [A, B, C, D] parameters. Similarly, the [Z], [Y], [ABCD] and [S] parameters are also converted among themselves [B.1, B.3, B.5].

      

      3.2.2 De‐Embedding of True S‐Parameters

A transmission line section could be treated as a device and its performance can be evaluated by using a VNA. The device is connected to the VNA through connecting cables and connectors. The S‐parameters of a device is measured at