S Parameters and the Scattering Matrix.
While Z and Y parameters can be useful descriptions for networks, S and ABCD parameters are even more widely used in microwave circuit work. We’ll begin with the scattering (or S) parameters. Consider again the multi-port network from the last lecture, which is connected to N transmission lines as:
Rather than focusing on the total voltages and currents (i.e., the sum of “+” and “-” waves) at the terminal planes t _{ 1} ,…., t_{n} , the S parameters are formed from ratios of reflected and incident voltage wave amplitudes. When the characteristic impedances of all TLs connected to the network are the same (as is the case for the network shown above), then the S parameters are defined as
Or
[V^{-}] = [s] . [V^{+}]
Where[s]is called the scattering matrix. As we defined in the last lecture, the terminal planes are the “phase = 0” planes at each port. That is, with
V_{n} (Z_{n} ) = V_{n}^{+ } e ^{-jbn} (z_{n} -t_{n} ) + V_{n} ^{-} e ^{-jbn (zn -tn )} n = 1,……N then at the terminal plane t_{n}
V_{n} (Z _{n} = t_{n}) º V_{n}^{+} + V_{n} ^{-}
Each S parameter in (1) can be computed as
Notice in this expression that the wave amplitude ratio is defined “from” port j “to” port i:
S_{ij}
Let’s take a close look at this definition (2). Imagine we have a two-port network:
Then, for example,
Simple enough, but how do we make V_{2} ^{+ } = 0? This requires that:
1. There is no source on the port-2 side of the network, and
2. Port 2 is matched so there are no reflections from this port.
Consequently, with V_{2} ^{+ } = 0 Þ S_{11} = T_{11} , which is the reflection coefficient at port 1. Next, consider
Again, with a matched load at port 2 so that V_{2} ^{+ } = 0, then
S_{21} = T_{21}
which is the transmission coefficient from port 1 to port 2. It is very important to realize it is a mistake to say S_{11} is the reflection coefficient at port 1. Actually, S_{11} is this reflection coefficient only when V_{2} ^{+ } = 0 .
As we’ll see in the following example, if port 1 is not matched, then the reflection coefficient at port 1 will generally depend not only on S_{11} , but also all other S parameters and the load. An advantage of using S parameters compared to others is that matched loads are used for terminating the ports rather than opens and shorts. In some circuits this difference is critical. For example, with transistor amplifiers a nearly matched load may be necessary for the amplifier to operate correctly, whereas an open or short load may render the amplifier nonfunctional.
Example N15.1. (Similar to text example 4.5.) A two-port network has the following S matrix referred to some system impedance Z_{0}:
If a short circuit is connected to port 2, what is the resulting return loss at port 1? From the definition (1)
we find that
V_{1} ^{-} = S_{11} V_{1} ^{+} + S_{12} V_{2} ^{+}
V_{2} ^{-} = S_{21} V_{1} ^{+} + S_{22} V_{2} ^{+}
How can we incorporate the short circuit load into these equations? Start with TLs connected to both ports as:
From this circuit, we see that
It is critical to realize this definition for T_{L} is the inverse of what you may have first thought, because of the assumed direction of the “incident” waves for S parameters: which is always into the port. So, for a short circuit load, T_{L} = - 1 Þ V_{2} ^{+} = - V_{2} ^{-} Using this result in (4) we find
V_{1} ^{-} = S_{11} V_{1} ^{+} + S_{12} V_{2} ^{-}
V_{2} ^{-} = S_{21} V_{1} ^{+} + S_{22} V_{2} ^{-}
Our desired result is the input reflection coefficient T =V V_{1} ^{-} /V_{1} ^{+} .Rearranging (6), we obtain
Substituting this into (5) yields
So that
This is the input reflection coefficient expression for a two-port terminated in a short circuit. Substituting the numerical values for the S parameters in (3) we find T = 0.633, so that RL = -20log_{10} T = 3.97 dB.
Again, it is crucial to realize that the input reflection coefficient of a two-port network is generally not S_{11} ! In the above example, S_{11} = 0.1 while T = 0.633. Remember that T_{11} = S only when all other ports are terminated in matched loads. By similar reasoning, S_{ij} = T_{ij} , i ¹ j , is valid only when all ports are matched. As we vary load or source impedances connected to a two-port network, the S parameters do not change. However, the reflection and transmission coefficients will generally change