S11, S21, S12 and S22 Explained for a Two-Port

S-parameters describe how waves travel through a network. For a two-port there are four of them: S11, S21, S12 and S22. Once you know how the names are built, you can read any two-port plot, whether it shows a filter, an amplifier or a cable.

What a two-port is

A two-port is any device with one connection on each side: a cable, a filter, an attenuator, an amplifier. Call the connections port 1 and port 2. You can send a wave into either port, and a wave can come out of either port.

Four things can happen, and each has its own S-parameter. The figure shows the box and the four paths.

How the numbering works

In Sxy, the first number is the port where the wave comes out. The second number is the port where it was applied. The order looks backwards at first, but it matches the matrix notation:

Here a is the wave going into a port and b is the wave coming out. So S21 is the wave leaving port 2 for a wave applied at port 1. S12 is the wave leaving port 1 for a wave applied at port 2.

The definition has one condition. When you measure S21, the other port must be terminated in the reference impedance, so nothing reflects back into it. Otherwise the wave at port 2 is not the clean response of the device.

Reflection and transmission

The four parameters fall into two groups.

• S11 and S22 are reflections. S11 is the part of the wave sent into port 1 that comes back out of port 1. S22 is the same thing at port 2. They tell you how well each port is matched to the reference impedance.
• S21 and S12 are transmission. S21 is the forward path, port 1 to port 2. S12 is the reverse path, port 2 to port 1.

All four are complex numbers. They have a magnitude and a phase. Most plots show only the magnitude in dB, and the phase is a second trace.

Magnitude in dB

Magnitudes are usually shown in dB as 20*log10(|S|):

Two points are worth remembering:

• A magnitude of 1 is 0 dB. The wave passes, or reflects, in full.
• A magnitude of 0.5 is about -6.02 dB.

The factor is 20 because S is a wave amplitude, not a power. Power goes as the square of the amplitude, so the fraction of power is |S| squared.

Worked example: a cable

Suppose a cable has |S21| = 0.5 and |S11| = 0.1 at some frequency. These numbers are chosen for the example, not measured.

1. Transmission: 20*log10(0.5) is about -6.02 dB.
2. Power through: 0.5 squared is 0.25, so 25 percent of the power reaches port 2.
3. Reflection: 20*log10(0.1) is -20 dB.
4. Power reflected: 0.1 squared is 0.01, so 1 percent comes back to port 1.
5. The rest, 100 - 25 - 1 = 74 percent, is absorbed in the cable as heat.

Because a cable is reciprocal, S12 is also 0.5, or -6.02 dB.

Reading a filter, an amplifier and a cable

Cable. It is passive and reciprocal, so S21 equals S12. A good cable has S21 close to 0 dB and S11 and S22 very low. S21 gets more negative as frequency rises, because the loss grows. The plot is a gentle downward slope, the same in both directions.

Filter. A plain passive filter is also reciprocal, so S21 equals S12. In the passband, S21 stays near 0 dB and S11 is low. In the stopband, S21 drops steeply. There the filter does not absorb most of the power. It reflects it, so S11 rises toward 0 dB. The next figure shows this for a low-pass sample filter. The curves are computed from the circuit model, not measured data.

Amplifier. Here the four parameters are all different:

• S21 is the gain. A value above 0 dB means the device amplifies.
• S12 is the reverse transmission, and it is small. An amplifier does not pass signals backward well, so S12 is not equal to S21.
• S11 and S22 show the input and output match. They usually differ from each other.

Whenever S21 and S12 differ, the device is not reciprocal. An amplifier is the typical case.

The reference impedance

S-parameters are defined for a reference impedance, normally 50 ohm. The "no reflection" condition above means "terminated in that impedance". A device that is well matched to 50 ohm shows a low S11. The same device referenced to another impedance gives different numbers, because the reflection is measured against a different standard.

To see the four traces of your own file, open it in a viewer. The page How to Open an S2P (Touchstone) File: 6 Ways and a Free Viewer shows how.

In plain words · for students and beginners

S-parameters in plain words

An everyday analogy

Think of a window in a room. You shine a torch at the glass from outside. Some light goes through, some bounces back at you, and some is absorbed by the glass.

• The light that bounces back is the reflection. That is S11.
• The light that reaches the inside is the transmission. That is S21.

Now go inside and shine the torch outward. You get two more numbers, S22 and S12. For plain glass the two directions behave the same. That is what "reciprocal" means.

The analogy breaks down for an amplifier. A window never makes light brighter, and an amplifier does. An amplifier also works strongly in one direction and weakly in the other.

The cable example in simple steps

Use the same cable as before. The numbers are made up for the example.

1. Send a signal into port 1.
2. Half of the wave amplitude comes out at port 2. That is |S21| = 0.5, which is about -6.02 dB.
3. A tenth of the amplitude bounces back. That is |S11| = 0.1, which is -20 dB.
4. Power is the amplitude squared: 25 percent comes out, 1 percent bounces back, and the other 74 percent turns to heat.
5. Send a signal in at port 2 instead. Because a cable is reciprocal, S12 is also 0.5.

Now look at the curves for a simple filter. The upper curve is S21, which stays high where the filter passes the signal and falls where it blocks it. The lower curve is S11, which does the opposite.

Mistakes beginners make

• Reading the numbers backwards. S21 is not "from port 2 to port 1". It is the wave leaving port 2 for a wave applied at port 1.
• Expecting S21 above 0 dB from a passive part. A cable or a plain filter can only lose signal, so its S21 is 0 dB or below.
• Forgetting the 50 ohm. The numbers belong to a reference impedance. Change it and the numbers change. Reference impedance means the standard resistance the measurement is compared against.
• Mixing up dB and plain numbers. A magnitude of 0.5 is about -6.02 dB, not -0.5 dB.

Remember

• In Sxy, x is where the wave comes out and y is where it was applied, so S21 is the forward transmission.
• S11 and S22 are reflections, S21 and S12 are transmission, and a cable or plain filter has S21 equal to S12.
• Magnitude in dB is 20*log10(|S|): 1 is 0 dB, 0.5 is about -6.02 dB, and everything is relative to a reference impedance, normally 50 ohm.

Related pages on RF View

• How to Open an S2P (Touchstone) File: 6 Ways and a Free Viewer
• Online S2P and Touchstone File Viewer