S-parameters

In eletronic circuits, (complex) Scattering parameters or s-parameters describe the scattering and reflection of traveling waves when a (linear) network is inserted into a transmission line. The use of s-parameers is typical but not limited to high-frequency applications, and are often measured and displayed as functions of frequency, e.g. using a Smith chart.

For each port, the applied (incident) and reflected wave are measured. When the incident wave travels through the network, its value is multiplied (i.e. is gain and phase are changed) by the scattering parameter.

Definition

s-parameters completely describe the behavior of a linear device. The individual parameters, dimensionless, complex numbers normally expressed as magnitude and phase. For a 2-port network, the most common case, they are

S₁₁: input reflection coefficient

S₂₁: forward transmission coefficient (gain)

S₁₂: reverse transmission coefficient (isolation)

S₂₂: output reflection coefficient

Note: Pronounce as "s-one-one" etc., not "s-eleven".

For low frequencies, H-, Y- and Z-Parameters are commonly used to describe n-port networks. However, determining either requires current measurements, which are not easily performed at higher frequencies, and require ports to be open- or short-circuited, which is also impractical at RF.

In contrast, S-parameters are measured with all ports terminated by their nominal impedance, most often 50 Ohms. The use of directional couplers permits measurements at very high frequencies.

Discussion

The S-parameters are measured by sending a signal (per frequency selected) into the device and detecting what returns as a reflex. The output (and other ports, if any) of the device is/are terminated with a 50-ohm resistor during the measurement. Only in one special case will there be no reflected signal. This is when the device has exactly the same impedance (50 Ω) as the source of the signal. Otherwise there will be a reflex.

If the test object it replaced by a perfect open (or short) circuit, the reflex will be total. The “Reflection Coefficient” will be 1, unity. In the matched case it would be 0 as there is no reflex. The difference between the S-parameter for an open and a short is that the phase between outgoing signal and reflected signal is 180° in a short and 0° in an open.

We realize now that if there is a transmission line (such as a coaxial cable, a wave guide, a twin-lead or other) between the point where we make our measurement and the actual device (or the short/open end of the line) we can get any phase we want. It depends on the length of the cable and the frequency. For example, at 100 MHz and with the very common RG-58 cable, one meter of the cable is ½ wavelength (l), so the reflex has to go twice (back and forth) in this cable. The reflected wave will look just like the injected signal! What is at the end of the cable will also appear to be on our end of the cable! The phase will rotate one full turn, 360°. This is the same as 0°, the phase of the initial signal. But for another frequency, or another cable length, this is no longer true. Intuitively we realize that we cannot match a bad load to a source with any length of cable of the same Characteristic Impedance.

The tested device may have one, two (or more) “ports”. A port is the connection to the device, input, output or whatever it is. The input is usually called “Port 1”. The output (if any) is called “Port 2”. Signals reflected from the input are called S11. This is interpreted as “Scattering on port 1 resulting from a signal on port 1” the reflex from the input port in other words. So signals on the output, resulting from signals applied to port 1 are called S21. This can be the gain of an amplifier, the insertion loss in a filter or a relay or the isolation of something not supposed to let signals through. In the same spirit, S12 is the “Reverse Gain” if an amplifier. In a filter, attenuator, a relay or any other passive device, the direction of the signal will not matter, so S21 = S12 and S11 can be identical to S22 if it is a symmetrical device.

The S-parameters are presented as a table of numbers. Here, as an example, are a few lines from the table of values for a fairly common and simple amplifier (MAR-6):

 Freq        S11	         S21            S12             S22
 MHz	  dB	Ang	  dB	Ang	  dB	Ang	dB	Ang
 100	-27.96	 171	 20.1	171	-22.50	 5	-27.96	 -30
 500	-26.02	-105	 18.7	138	-21.30	21	-20.00	-104

So, fed from a 50Ω source at 100 MHz, we can see that the input of this amplifier reflects 0.04 of the voltage, or –28 dB. This is a very good match. 998.3/1000 of the power is absorbed. The angle of the reflex is 171 degrees, so the small reflex is “towards a short circuit” rather than towards an open circuit. A few Ω below 50.

The gain of the amplifier, 20.1 dB (10X in voltage or 100X in power) is also 171°. That is: the phase of the output signal is almost inverted, rotated 180°. This is normal and good.

S12 is the “Isolation”. The 12 means: signal appearing on port 1 from signal applied to port 2. In an amplifier it is sometimes also called “Reverse Gain” and is of no use. It can even create problems with the stability. Here it is small. A signal of strength 1 applied to the output will result in a signal strength (in voltage) of 0.075 on the input. The attenuation is 22.5 dB, more than the gain in the forward direction. That is good too. The angle of the signal, 5°, is about the same as the applied, so it is all just a small leak!

S22 is also called the “Output Match”. It is very good here. A signal (from the 50 Ω source) applied to the output reflects very weakly. Only –28 dB comes back and the angle is about the same as the applied signal, 30°. A good output match is a good sign for two reasons: 1) the power from the amplifier is “well taken care of” and travels down the cable with very small loss from reflections. 2) If the signal is reflected back into the cable from the other end, this reflex will be almost totally absorbed by the amplifier output. One say that the system is well “Back-terminated”. If the application for this amplifier is as an antenna amplifier, there will be no ghost images from reflected signals that go back and forth in the cable.

The S-parameters are an extremely useful method for describing devices used at Radio and Microwave Frequencies. There is no real limitation to these frequencies, but there is where they are most commonly used.

The scattering, the Reflection Coefficient (upper case lambda, looks like an upside down L) and its angle, can be plotted in a very famous circular diagram, the Smith Chart (after an Engineer at Bell Laboratories, Phillip H. Smith, who developed this during the 1930-s) in which parameters can be plotted and calculations done for matching and others. This chart is essential in understanding the nature of transmission lines.

A few amazing examples of the application of S-parameters:

Imagine a filter circuit! The filter is “sabotaged” in that one of the capacitors has the wrong value. It works somewhat, but not well. The filter is measured up (with a “Vector Network analyzer”, a VNA) and a table with the S-parameters is generated. A circuit is created on a computer where the filter appears as a block, characterized by its S-parameters. The performance is flawed, as we could see on the network analyzer. Now, on the computer, we can start adding external components that will make the filter better. If we are lucky the flawed value is on a port (so we can get to it directly) and if the problem is, for example, lack of capacitance we can add capacitance on the port and get the performance back! If the faulty component is not available on the outside, possibly an additional network can be created that fixes the filter. Of course, the right thing to do is to fix the filter itself. But the S-parameters are so good a description of the real parts that even a faulty part can be “repaired” on the computer!

Imagine an unknown coaxial cable is at hand! We measured the length to 20 meter. We do not know its characteristic impedance and its attenuation at 100 MHz.

Solution: connect it to a VNA and sweep it in frequency. The reflex from the open (or shorted) end will come back as a total reflex. As the swept frequency changes the vector that can describe the Reflection Coefficient vector will rotate. For a very low frequency, and an open cable end, it starts at 0° (at 9 o’clock) and a full length as the cable barely has any attenuation (loss) yet. As the frequency increases the vector will rotate clockwise and when the cable is ½ wavelength long the vector is back at 0°. It is a little bit shorter than 1 as the wave has been attenuated when traveling both ways. At 3/2 wavelengths it is back again at the same angle, and even a little bit shorter. As frequency goes up the tip of the vector will describe a spiral, passing 9 o’clock every odd ½ wavelength. The position of the center of the spiral is the Characteristic Impedance of the cable! If it is 50Ω the spiral is centered on the screen. If it is 75 Ω the spiral is off to the right, at 1.5 on the Real axle on the Smith Chart.

Setting the frequency to 100 MHz, or reading at this frequency, the length of the vector will show twice the attenuation of the cable! On a rectangular VNA display we can read this directly. If it reads 4 dB, the length of cable has a 2 dB loss (each way). This cable could also be described by S-parameters if one so prefer.

Imagine an antenna connected to a VNA! If the antenna is any good at radiating signal at the frequency of interest, the S11 will read about 0.2 or less. Not much energy comes back. 0.1 is very good for an antenna. If someone walks by, near the antenna, some of the energy is reflected back into the antenna, into the cable, back to the VNA! The VNA can be normalized to the small mismatch the antenna has, making it look like a “Perfect Match”. Differences as small as –80 dB (1/100 millionths of the power!) can be now be detected. A good, but expensive, burglar alarm! A directional antenna, like a TV antenna for UHF, can cover a “corridor” of several hundred meters.

External links

• Application Note 154, S-Parameter Design, Agilent
• AN95-1, S-Parameter Techniques, Agilent