A Quick Look at S-Parameters

Back in the late 1970s I was an Electronics Technician with TelePrompTer, one of the country’s largest cable operators at the time. Part of my job was equipment testing and repair, some of which involved using fairly simple bench sweep equipment to measure bandwidth, insertion loss, gain, flatness, and so on. In the 1980s I ran Jones Intercable’s corporate engineering lab, evaluating various products for the company. We acquired a vector network analyzer (VNA), which can be thought of as a very sophisticated bench sweep (it actually does a lot more). Operating the VNA meant learning something new to me at the time: scattering parameters, more commonly called S-parameters.

Unless you’re involved with the use of specialized test equipment such as VNAs, S-parameters are likely something you’ve never heard of. At least, perhaps, until recently. In the world of full duplex (FDX) DOCSIS, the S-parameter “S₁₁” is commonly used when discussing an FDX node’s echo cancellation operation. I’ve seen some people refer to S₁₁ as return loss, which it’s not. Return loss can be derived from S₁₁ (more on this later), but the two aren’t quite the same thing. For some background on return loss, see my Broadband Library article on the subject at https://broadbandlibrary.com/return-loss/. So, what are S-parameters? Read on for a high-level overview.

Consider a component or device being evaluated as a “network” with some number of ports N, and a characteristic impedance Z₀. For instance, a terminator can be considered a one-port network, as shown in Figure 1.

Figure 1. Cable equipment chassis terminator represented as a one-port network. 

 

An in-line attenuator (or amplifier, tap, etc.) can be considered a two-port network, as shown in Figure 2.

Figure 2. In-line attenuator represented as a two-port network.

 

Figure 3 shows a line extender amplifier as a two-port network.

Figure 3. Amplifier represented as a two-port network.

 

Among the various metrics that can be used to characterize N-port networks are scattering parameters. According to Wikipedia, S-parameters “…describe the electrical behavior of linear electrical networks when undergoing various steady state stimuli by electrical signals.” S-parameters are a simplified representation of a black box network, and are complex numbers — that is, S-parameters have a magnitude and phase component. They are expressed in the format S_mn where “m” is the output port number during the measurement and “n” is the input port number during the measurement. For a two-port network, either port can be an input or output port, depending on the measurement; so, in most cases it’s better to refer to port numbers. The “n” subscript also represents the port to which the incident (test) signal is applied. The remainder of this discussion focuses on two-port networks.

Assume that the performance of a two-port network such as an attenuator is being characterized; see Figure 4.

Figure 4. Testing the two-port network from Port 1 to Port 2 gives the S-parameter S₂₁.

 

A test signal (“Incident”) is applied to Port 1, and that signal (“Transmitted”) is measured at Port 2 after it passes through the device. From an S-parameter perspective this gives us S₂₁, the ratio of transmitted voltage E_Transmitted to incident voltage E_Incident, called the forward voltage gain or transmission coefficient. As you look at Figure 4, the “21” in S₂₁ indicates the voltage is applied at Port 1 (the “1” in “21”) and is measured at Port 2 (the “2” in “21”). An important point: S-parameters are frequency dependent — that is, the ratio that describes, say, S₂₁, is valid only for a given frequency. As well, S-parameters are referenced at the measurement or interface plane of a network; examples of that plane are Port 1 and Port 2 in Figure 4 (rather than at the test equipment making the measurement).

While the incident signal is being applied to Port 1, we can also measure any reflection from that port. The reflection is a combination of reflections from Port 1’s connector and components inside of the two-port network. Here, the S-parameter is S₁₁, the ratio of reflected voltage E_Reflected to incident voltage E_Incident, and is called the voltage reflection coefficient for Port 1. See Figure 5.

Figure 5. Measuring the reflection from Port 1 gives the S-parameter S₁₁.

 

Note:  When performing a measurement of S-parameters, the port(s) other than the one(s) being measured should be terminated in the network’s characteristic impedance (typically 75 ohms for cable systems). An example is shown in Figure 6 for the case of a reflection measurement at Port 1, with Port 2 terminated.

Figure 6. It is common when measuring S₁₁ to terminate Port 2 in an impedance equal to Z₀.

 

Next, apply a test signal to Port 2, and measure at Port 1. This gives us the S-parameter S₁₂, the ratio of E_Transmitted to E_Incident, and is the reverse voltage gain or transmission coefficient. See Figure 7.

Figure 7. Testing the network from Port 2 to Port 1 gives the S-parameter S₁₂.

 

A reflection measurement can be made at Port 2, as shown in Figure 8.

Figure 8. Measuring the reflection from Port 2 gives the S-parameter S₂₂. In many cases Port 1 would be terminated in an impedance equal to Z₀ while performing this measurement (terminator not shown). 

 

This gives us the S-parameter S₂₂, or Port 2’s voltage reflection coefficient. Here, too, the reflection is a combination of reflections, in this case from Port 2’s connector and components inside of the two-port network.

In an operational FDX DOCSIS network, the FDX node’s echo cancellation circuitry characterizes the network from the perspective of S₁₁, similar to the example shown in Figure 9.

Figure 9. FDX DOCSIS node S₁₁ characterization of the cable network.

 

Table 1 summarizes the S-parameters just discussed.

As mentioned previously, for S₁₁ and S₂₂, the voltage reflection coefficient is the ratio of reflected voltage E_Reflected to the incident voltage E_Incident. For S₁₂ and S₂₁, the transmission coefficient is the ratio of transmitted voltage E_Transmitted to incident voltage E_Incident.

Figure 10. Relationships of E_Incident, E_Transmitted, and E_Reflected when measuring from Port 1 to Port 2.

 

Figure 10 shows the relationships of E_Incident, E_Reflected, and E_Transmitted when measuring from Port 1 to Port 2. Here, E_Transmitted/E_Incident = S₂₁ and E_Reflected/ E_Incident = S₁₁.

Going the other direction, from Port 2 to Port 1, E_Transmitted/E_Incident = S₁₂ and E_Reflected/E_Incident = S₂₂.

See Figure 11.

Figure 11. Relationships of E_Incident, E_Transmitted, and E_Reflected when measuring from Port 2 to Port 1.

 

Cable operators are usually more familiar with characteristics such as gain, insertion loss, and return loss, all expressed in decibels. The following formulas are used to derive those metrics from S-parameters (referenced to a two-port network, and assuming Port 1 is the input port and Port 2 is the output port). Voltage standing wave ratio (VSWR) and impedance match also can be derived from S-parameters; see the Wikipedia scattering parameters article referenced below for more information.

 

Input return loss in decibels  R_in = –20log₁₀|S₁₁|

 

Output return loss in decibels  R_out = –20log₁₀|S₂₂|

 

Gain in decibels  G_dB = 20log₁₀|S₂₁|

 

Insertion loss in decibels  L_dB = –20log₁₀|S₂₁|

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If you’d like to learn more about S-parameters, I encourage you to refer to the following.

Wikipedia: Scattering parameters

https://en.wikipedia.org/wiki/Scattering_parameters

Microwaves101.com: S-parameters

https://www.microwaves101.com/encyclopedias/s-parameters

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Ron Hranac

Technical Marketing Engineer,
Cisco Systems
rhranacj@cisco.com

Ron Hranac, a 46-year veteran of the cable industry, is TME for Cisco’s Cable Access Business Unit. A Fellow Member of SCTE and co-founder and Associate Board Member of the organization’s Rocky Mountain Chapter, Ron was inducted into the Society’s Hall of Fame in 2010, is a co-recipient of the Chairman’s Award, an SCTE Member of the Year, and is a member of the Cable TV Pioneers Class of ’97. He received the Society’s Excellence in Standards award at Cable-Tec Expo 2016. He has published hundreds of articles and papers, and has been a speaker at numerous international, national, regional, and local conferences and seminars.

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