Handbook of Microwave Component Measurements. Joel P. Dunsmore

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(1.21)

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 S_nn parameters are reflection coefficients and are directly related to the DUT port input impedance and S_mn 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 Z_L 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

(1.24)

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 Γ_In 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, Z_L, as

(1.27)

where Γ_L computed as in Eq. (1.24) is

      (1.28)

      or in the case of a 2‐port network terminated by an arbitrary load then

      (1.29)

      Similarly, the output impedance of a network that is sourced from an arbitrary source impedance is

(1.30)

      Another common term for the input impedance is the voltage standing wave ratio, called VSWR (also simply called SWR), and it represents the ratio of maximum voltage to minimum voltage that one would measure along a Z₀ transmission line terminated in some arbitrary load impedance. It can be shown that this ratio can be defined in terms of the S‐parameters of the network as

      (1.31)

If the network is terminated in its reference impedance, then Γ₁ becomes S₁₁. Another common term used to represent the input impedance is the reflection coefficient, ρ_In, where

      (1.32)

      It's also common to write

      (1.33)

      Another term related to the input impedance is return loss, which is alternatively defined as

      (1.34)

      with the second definition being most properly correct, as loss is defined to be positive in the case where a reflected signal is smaller than the incident signal. But, in many cases, the former definition is more commonly used; the microwave engineer must simply refer to the context of the use to determine the proper meaning of the sign. Thus, an antenna with 14 dB return loss would be understood to have a reflection coefficient of 0.2, and the value displayed on a measurement instrument might read −14 dB.

      For transmission measurements, the figure of merit is often gain or insertion loss (sometimes called isolation when the loss is very high). Typically this is expressed in dB, and similarly to return loss, it is often referred to as a positive number. Thus

(1.35)

      Insertion loss or isolation is