### Section 3: S-Parameters

In the Lesson 1 movies, we saw short pulses bouncing along the lines.  These pulses would sum constructively or destructively when they hit other pulses.  When we saw movies of DC or digital voltage waves, it was harder to see the actual voltage waves.  We saw the sum of the voltage waves along the transmission lines.  But we actually have discrete incident (transmitted) and reflected waves.

#### Sum and Reflected Waves

Click the button in the lower right to make this video full screen.

S-Parameters are relationships of the voltage waves entering and leaving the system. They are the ratios of reflected and incident waves, a sort of gain- how much leaves the system relative to how much went in.  Recall the classic voltage waves traveling down a transmission line:

Figure 2.3.1: Incident and reflected waves

V1+ is the wave going into this system, and V1- is the wave going out of the system. In the movie above, the red plot was V1-, and the blue plot was the sum of V1- plus V1+.  It turns out that the ratio of the two voltage waves is related to the load (ZL) and transmission line (Z0) impedance.  We are going to define the ratio of the voltage waves to be a "scattering parameter", or S-Parameter.

S-Parameters are defined as relationships between the incident and the reflected waves. The nomenclature is: Sri, where 'r' is the port where energy is coming out of the system and 'i' is the port where energy is entering the system. For a 1-port system, there is only one S-Parameter: S11.  Energy enters and leaves out of port 1, the only port.

Things get more interesting for a two-port network. For a two port systems, there are four S-Parameters: S11, S21, S12, and S22. The following definitions and points extend for whenever you have more than one port. While perhaps not intuitive, the voltage wave leaving the system (V1- or V2-) is always referred to as the "reflected" wave, even if the V2- wave is the result of (was excited by) V1+.  Since V1+ sends a wave to the right (as in Figure 2.3.2), and the V2 wave is also going to the right, it might not be obvious to call that a reflection- but that's how it is defined. Consider the following two-port network with four relevant voltage waves.

Figure 2.3.2: Two port network's incident and reflected waves

As in the one-port case, the incident and reflected voltage waves are related to the S-Parameter.

It's not too difficult to see what we need to do to define the S-Parameter: one port has to be driven (for example, V1+ = 1) and the rest of the ports need to be turned off (for example, V2+ = 0).  But, not only can we not excite any voltage waves on port 2, we can't let any wave that leaves port 2 reflect back- port 2 needs to be perfectly terminated.  Recall in our movies in Lesson 1, if the right side is not loaded correctly, we get a lot of voltage bounces.  But when we had our 50ohm transmission line terminated with 50ohms, we got the same effect as we are looking for when we say V2+ = 0.  So, port 2 must be matched- typically to 50ohms.

At this point we have defined the relationship between the incident and reflected voltage waves to a single S-Parameter. The actual voltage is the sum of the incident and reflected voltage, V1 = V1++ V1-, as in Figure 2.3.3 (as copied from Lesson 1 Section 4).

Figure 2.3.3: Wave Bounce Diagram showing voltage sums at transitions

Main Point

S-Parameters are relationships between voltage waves entering and leaving the circuit. S-Parameters are individually defined with one port driven, and all other ports perfectly terminated. Driving more than one port? S-Parameters can't be applied directly. Port has some mismatch? Expect to see changes in system gain and match. Fortunately, the relationships aren't hard to derive (or find, down below).