of the network. It is a “nominal” impedance; that is, it is the impedance that we “name” when we are determining the S‐parameters, but it need not be associated with any impedance in the circuit. Thus, a 50 Ω test system can easily measure and display S‐parameters for a 75 Ω device, referenced to 75 Ω.
The etymology of the term reflected derives from optics and refers to light reflecting off a lens or other object with an index of refraction different from air, whereas it appears that the genesis for the scattering or S‐matrix was derived in the study of particle physics, from the concept of wavelike particles scattering off crystals. In microwave work, scattering or S‐parameters are defined to relate the independent incident waves to the dependent waves; for a 2‐port network they become
which can be placed in matrix form as
(1.18)
where a's represent the incident power at each port, that is, the power flowing into the port, and b's represent the scattered power, that is, the power reflected or emanating from each port. For more than two ports, the matrix can be generalized to
(1.19)
From Eq. (1.17) it is clear that it takes four parameters to relate the incident waves to the reflected waves, but Eq. (1.17) provides only two equations. As a consequence, solving for the S‐parameters of a network requires that at least two sets of linearly independent conditions for a1 and a2 be applied, and the most common set is one where first a2 is set to zero, the resulting b waves are measured, and then a1 is set to zero, and finally a second set of b waves are measured. This yields
which is the most common expression of S‐parameter values as a function of a and b waves, and often the only one given for their definition. However, there is nothing in the definition of S‐parameters that requires one or the other incident signals to be zero, and it would be just as valid to define them in terms of two sets of incident signals, an and
From Eq. (1.21) one sees that S‐parameters are in general defined for a pair of stimulus drives. This will become quite important in more advanced measurements and in the actual realization of the measurement of S‐parameters, because in practice it is not possible to make the incident signal go to zero because of mismatches in the measurement system.
These definitions naturally lead to the concept that Snn parameters are reflection coefficients and are directly related to the DUT port input impedance and Smn parameters are transmission coefficients and are directly related to the DUT gain or loss from one port to another.
Now that the S‐parameters are defined, they can be related to common terms used in the industry. Consider the circuit of Figure 1.3, where the load impedance ZL may be arbitrary and the source impedance is the reference impedance.
Figure 1.3 1‐port network.
From inspection one can see that
(1.22)
which is substituted into Eq. (1.8) and Eq. (1.9), and from (1.15) one can directly compute a1 and b1 as
(1.23)
From here S11 can be derived from inspection as
It is common to refer to S11 informally as the input impedance of the network, where
(1.25)
This is clearly true for a 1‐port network and can be extended to a 2‐port or n‐port network if all the ports of the network are terminated in the reference impedance; but in general, one cannot say that S11 is the input impedance of a network without knowing the termination impedance of the network. This is a common mistake that is made with respect to determining the input impedance or S‐parameters of a network. S11 is defined for any terminations by Eq. (1.21), but it is the same as the input impedance of the network only under the condition that it is terminated in the reference impedance, thus satisfying the conditions for Eq. (1.20). Consider the network of Figure 1.2 where the load is not the reference impedance; as such, it is noted that a1 and b1 exist, but now Γ1 (also called ΓIn for a 2‐port network) is defined as
(1.26)
with the network terminated in an arbitrary impedance. As such, Γ1 represents the input impedance of a system comprised of the network and its terminating impedance. The important distinction is that S‐parameters of a network are invariant to the input of output terminations, providing they are defined to a consistent reference impedance, whereas the input impedance of a network depends upon the termination impedance at each of the other ports. The value of Γ1 of a 2‐port network can be directly computed from the S‐parameters and the terminating impedance, ZL, as
