S-parameters

S-parameter stands for Scattering Parameter. It has come to replace all kinds of measurement parameters, mostly Y and Z, characterizing transistors, from mid-1900.

Since about 1970 - 80 the S-parameters are the standard way to characterize not only transistors, but also all kinds of components in Radio Frequency and Microwave systems.

Computer programs can readily use S-parameters for to calculate how an entire assembly of components (each described by its S-parameters) will respond to signals.

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. The difference between the S-parameter for an open and a short is that the phase between outgoing signal and reflected signal is 0° in a short and 180° 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) 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. It will look just like if there was no cable! 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°. But for another frequency, or another cable length, this is not true. Intuitively we realize that we cannot match a bad load to a source with just a 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), but they are a bit confusing as the particular manufacturer has elected to show three of the four parameters with Magnitude (also called G with values between 0 and 1) and angle as is conventional, but also in dB. For the gain, S21 the Magnitude is not shown, but a 20 dB value is 10 times (voltage) of gain.

Freq	  	 S11		      S21			S12			S22

!MHz dB Mag Ang dB Ang dB Mag Ang dB Mag Ang 100 -27.96 0.04 171 20.1 171 -22.50 0.075 5 -27.96 0.04 -30 500 -26.02 0.05 -105 18.7 138 -21.30 0.086 21 -20.00 0.10 -104 … .. .

Normally the table would have 9 columns: Frequency and Magnitude and Angle for each of the four S-parameters in the following order: S11 S21 S12 and S22.

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 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 little losses 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. The system is well “Back-terminated”. If the application for this amplifier is as an antenna amplifier, there can be ghost images if the reflected signals 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 used.

The scattering, the Reflection Coefficient G 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 in the late 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 created. 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.

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). As the frequency increases the vector will rotate clockwise and when the cable is ½ l (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 ¾ l 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 l /4. 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 the signal at the frequency of interest, the S11 will read 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, so it looks like a “Perfect Match” and differences as small as –80 dB (1/100 millionths of the power!) can 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.