Читать книгу Introduction To Modern Planar Transmission Lines - Anand K. Verma - Страница 124

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].