where ΓL 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
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, ρIn, 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
Insertion loss or isolation is defined as
(1.36)
Again, the microwave engineer will need to use the context of the discussion to understand that a device with 40 dB isolation will show on an instrument display as −40 dB, due to the instrument using the evaluation of Eq. (1.35).
Notice that in the return loss, gain, and insertion loss equations, the dB value is given by the formula 20log10(|Snm|), and this is often a source of confusion because common engineering use of dB has the computation as XdB = 10log10(X). This apparent inconsistency comes from the desire to have power gain when expressed in dB be equal to voltage gain, also expressed in dB. In a device sourced from a Z0 source and terminated in a Z0 load, the power gain is defined as the power delivered to the load relative to the power delivered from the source, and the gain is
The power from the source is the incident power |a1|2, and the power delivered to the load is |b2|2. The S‐parameter gain is S21 and in a matched source and load situation is simply
(1.38)
So computing power gain as in Eq. (1.37) and converting to dB yields the familiar formula
(1.39)
A few more comments on power are appropriate, as power has several common meanings that can be confused if not used carefully. For any given source, as shown in Figure 1.1, there exists a load for which the maximum power of the source may be delivered to that load. This maximum power occurs when the impedance of the load is equal to the conjugate of the impedance of the source, and the maximum power delivered is
But it is instructive to note that the maximum power as defined in Eq. (1.40) is the same as |a1|2 provided the source impedance is real and equals the reference impedance; thus, the incident power from a Z0 source is always the maximum power that can be delivered to a load. The actual power delivered to the load can be defined in terms of a and b waves as well.
(1.41)
If one considers a passive two‐port network and conservation of energy, power delivered to the load must be less than or equal to the power incident on the network minus the power reflected, or in terms of S‐parameters
(1.42)
which leads the well‐known formula for a lossless network
(1.43)
1.3.2 Phase Response of Networks
While most of the discussion thus far about S‐parameters refers to powers, including incident, reflected, and delivered to the load, the S‐parameters are truly complex numbers and contain both a magnitude and phase component. For reflection measurements, the phase component is critically important and provides insight into the input elements of the network. These will be discussed in great detail as part of
