Introduction To Modern Planar Transmission Lines. Anand K. Verma

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.

      Example 3.8

Determine the S‐parameters and return loss of a 2‐port network with arbitrary termination shown in Fig (3.14).

      Solution

      The 2‐port network (device) is connected to a source at the port‐1 and a load Z_L at the port‐2. The source has voltage V_g with internal impedance Z_g. The network scattering parameters‐[S] are computed under the matched condition. The characteristic impedance of the connecting line between the port‐1 and the source is Z₀₁, whereas the characteristic impedance of the connecting line between the port‐2 and the load is Z₀₂. The lengths of the connecting lines are zero. The reflection and transmission coefficients are to be determined at the input and output terminals. This is a practical problem for the measurement and simulation of the 2‐port network:

(i)

Figure 3.14 A two‐port network with arbitrary termination.

      Figure (3.14) shows that the power variable b₂ is the incident wave at the load Z_L and the power variable a₂ is the reflected wave from the load. Thus, the reflection coefficient at the load is

(ii)

From above equations (i) and (ii):

      (iii)

      On substituting b₂ from equation (b) in equation (a):

      (iv)

      The input reflection coefficient at the port‐1 is

      (3.1.56)

      The reflection coefficient Γ₁ is more than S₁₁ of the network. The mismatch at the load degrades the return loss (RL) of the network. It is given by

      (3.1.57)

      For the port 2 open‐circuited (Z_L → ∞), the waves get reflected in‐phase, i.e. Γ_L = 1, and for a short‐circuited load (Z_L = 0) the total reflection is out of phase, i.e. Γ_L = −1. If the network is terminated in a matched load (Z_L = Z₀₂), the incident waves are absorbed with Γ_L = 0 and Γ₁ = S₁₁. Likewise, the source reflection coefficient Γ_g could be defined at the input port‐1. Figure (3.14) again shows that b₁ is the incident wave on the internal impedance of the source Z_g and a₁ is the reflected wave from Z_g. Thus,

(v)

The output reflection coefficient Γ₂ at the port‐2 is obtained from equations (i) and (v):

      (3.1.58)

      Again under the matched condition (Z_g = Z₀₁) at the input port, Γ_g = 0. For most of the applications, 50 Ω system impedance is used, i.e. Z₀₁ = Z₀₂ = Z₀ = 50 Ω. For a 2‐port lossless network, we have the following expressions:

      However, for a reciprocal network S₁₂ = S₂₁. Thus, the above equations provide

      (3.1.59)

      The network also follows . The S‐parameters are complex quantities. The S‐parameters are written in the phasor form: , and S₁₂ = |S₁₂| e^{j ϕ}. From the above equation, the phase relation is obtained:

      (3.1.60)

      Therefore, once the complex S₁₁ and S₂₂ are measured, both the magnitude and phase of the S₂₁ are determined. However, usually, both S₁₁ and S₂₁ are obtained from a VNA and also from the circuit simulator or EM‐simulator. The magnitude of S₂₁ provides the insertion‐loss of the network and ϕ is the phase shift at the output of the network.