### Section 5: Interesting Scattering Parameter Properties

Scattering Parameters are very useful to RF designers.  For one, it turns out that it is easier to measure voltage waves than actual voltages in an RF circuit.  When you have pulses and sine waves going through an RF device, with lots of different reflections in the circuit due to mismatches, trying to describe what's going on with a single voltage or current becomes somewhat arbitrary.  S-Parameters also have many interesting properties which can help provide insight.

System of S-Parameter Equations

Consider a power divider- many houses have an RF power divider which splits the cable signal to different TVs around the house.  Consider specifically a four-way power divider, where port 1 is the input (the cable TV signal goes into here) and ports 2-5 are the output (these signals are sent to TVs around the house). If port 1 is driven, it makes sense that port 2 will have a quarter of the power coming out (as a voltage wave, -6dB less than the input). The actual S-Parameter equations for this system are:

Equation 2.5.1

Now, writing out equations like this can be cumbersome, so we are going to short hand these equations by putting the equivalent variables in matrix formulation.  If you aren't familiar with linear algebra, don't worry about the mechanics, just notice that the variables follow the same pattern... and we're going to be talking about patterns.  I assure you, these are equivalent sets of equations.

Equation 2.5.2

Reciprocal Devices
A broad class of components, such as our power divider, are known as reciprocal devices.  Reciprocal devices have no active devices (no transistors), no magnetic materials (don't worry how they work), and no plasmas.  If a device is reciprocal, it has the following property:

Equation 2.5.3

This means the device has symmetry to it.  Let's consider the above device where port 1 is driven by the cable signal from outside your house, and port 2 is one of four ports where power is split out to your TV.  Equation 2.5.3  says that if S21 is -6dB (25% of the power goes to port 2 from port 1), then S12 is also -6dB.  If you put 1W into port 1, then 0.25W will come out of port 2 (assuming everything is matched, no bad reflections).  If you put 1W into port 2, then 0.25W will come out of port 1 (assuming everything is matched, no bad reflections).  So, S21 = S12 for a reciprocal device.

Figure 2.5.1: Four-way splitter

If we consider a transistor amplifier, the device has gain.  The voltage wave coming out of port 2 will be larger than the voltage wave coming out of port 1, for example we described a 13dB gain amplifier above: S21 = 13dB.  However, if you connect the device backwards, if you drive power into the output, not much energy will come out of the input: S21 ≠ S12.  The S12 is often called the isolation of the amplifier, this is how well what the input is isolated from what happens on the output, and is commonly on the order of -20dB (or only 1% of the power gets through).

Figure 2.5.2: Amplifier with 13dB gain and -20dB isolation