Читать книгу Introduction To Modern Planar Transmission Lines - Anand K. Verma - Страница 112

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, the incident wave at the n^th port, x = − ℓ_n, after reflection from the port at x = 0, returns to x = − ℓ_n. In the process, it travels the electrical length 2θ_n. Similarly, if the wave is incident at port #1, located at x = − ℓ₁ and arrives at the port‐2, located at x = − ℓ₂; the electrical length traveled by the wave is θ₁ + θ₂ = β₁ ℓ₁ + β₂ ℓ₂, or 2θ₁, on the assumption that β₁ = β₂, and ℓ₁ = ℓ₂, i.e. the transmission lines connected at both the ports are identical. The measured or simulated scattering matrix [S^′] at the location x = − ℓ_n is related to the [S] parameters of the network by the following expression

(3.1.54)

The [S]‐parameter of the network is extracted from equation (3.1.54), as

(3.1.55)

For reducing the cascaded network to a single equivalent network, the [S] parameters cannot be cascaded like the [ABCD] parameters. The [ABCD] matrix is suitable for this purpose. However, it is not defined in terms of the power variables. Therefore, another suitable transmission matrix, called [T] matrix has been defined in terms of the power variables to cascade the microwave networks. The [S] matrix is easily converted to the [T] parameters [B.1, B.2–B.5, B.7, B.9].

The concept of the [S] matrix is used below to some simple, but useful circuits. These examples would help to appreciate the applications of the [S] parameters.