Two-Port Network Modeling and Transmission Line S-Parameters

Key Takeaways

• Measuring S-parameters is common in RF and microwave systems. These parameters can be measured with the help of network analyzers. 

• A transmission line is an example of a symmetrical two-port network, so interchanging port one and port two will not change the transmission properties.  

• Transmission line S-parameters are influenced by the characteristic impedance Z_c and propagation constant 𝛾.

 In RF circuits, transmission lines act as connectors

In RF circuits, transmission lines are laid with precise dimensions, thickness, spacing, and impedance matching so that electromagnetic signals are transmitted with minimum reflections and power loss. Two-port network modeling is used in transmission line analysis, where numerous parameters—such as Z-parameters, Y-parameters, ABCD parameters, H-parameters, and T-parameters—define the relationship between dependent and independent quantities.

Transmission line scattering parameters (S-parameters) are similar to the family of parameters described above. S-parameters are elements of the scattering matrix that are used to model the behavior of linear RF and microwave networks. In this article, we will take a closer look at transmission line S-parameters. 

A Generalized Two-Port Network Model of a Transmission Line and S-Parameters

Measuring S-parameters is common in RF and microwave systems. S-parameters can be directly measured with the help of network analyzers. In an S-parameter formulation, system inputs and outputs are replaced by incident waves and reflected waves. This approach is best suited for high-frequency RF and microwave applications. S-parameter transformations are based on the incidence-reflection concept.

The transmission lines in RF circuits that interconnect various components can be modeled as two-port networks with two incident waves, which are a₁ and a₂ in ports one and two (shown in the figure below). The reflected waves in ports one and two are given by b₁ and b₂, respectively. Usually, the incident and reflected waves are normalized with the characteristic impedance value. The general S-parameters of a two-port model of a transmission line can be written as: 

When the incident wave at port one and the output are matched, there are no reflections in the transmission line, so a2 becomes zero. Additionally:

• The parameter S11 gives the input reflection coefficient.
• S12 is the reverse transmission coefficient (from port two to port one).
• S21 is the forward transmission coefficient (from port one to port two).
• S22 is the output reflection coefficient.

Lossy and Lossless Transmission Line S-Parameters

A transmission line is an example of a symmetrical two-port network. The S-parameters of a symmetrical network share a unique relationship:

Interchanging port one and port two does not change any transmission line S-parameter properties. The factors that can change transmission line S-parameters are:

• Line geometry
• Characteristic impedance
• Frequency

When any of these factors change, the S-parameters and transmission line properties also change.

Transmission line S-parameters are influenced by the characteristic impedance Z_c and propagation constant 𝛾. The propagation constant 𝛾 and characteristic impedance Z_c are given by the following equation, where R, L, G, and C are the distributed elements of the transmission line of length, l:

As the transmission line is symmetrical and reciprocal, S₁₁=S₂₂ and S₁₂=S₂₁. The table below gives the S-parameters of the lossy and lossless transmission lines terminated by Z_L.     

This table shows the S-parameters of lossy and lossless transmission lines

Transmission Line S-Parameter Frequencies

Voltage and current are more like traveling waves at high frequencies, and approaching transmission line analysis and synthesis using transmission line S-parameters is more suitable at RF and microwave frequencies. Cadence’s software supports the analysis of RF circuits and transmission lines by supporting the calculation of S-parameters.  

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