Handbook of Microwave Component Measurements. Joel P. Dunsmore

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(|S_nm|), and this is often a source of confusion because common engineering use of dB has the computation as X_dB = 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

(1.37)

      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

(1.40)

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 Chapter 2, especially when referencing the Smith chart.

      For transmission measurements, the magnitude response is often the most cited value of a system, but in many communications systems, the phase response has taken on more importance. The phase response of a network is typically given by

      (1.44)

where the region of the arctangent is usually chosen to be ±180°. However, it is sometimes preferable to display the phase in absolute terms, such that there are no phase discontinuities in the displayed value. This is sometimes called the unwrapped phase, in which the particular cycle of the arctangent must be determined from the previous cycle, starting from the DC value. Thus, the unwrapped phase is uniquely defined for an S₂₁ response only when it includes all values down to zero frequency (DC).

The linearity of the phase response has consequences when looking at its effect on complex modulated signals. In particular, it is sometimes stated that linear networks cannot cause distortion, but this is true only of single‐frequency sinusoidal inputs. Linear networks can cause distortion in the envelope of complex modulated signals, even if the frequency response (the magnitude of S₂₁) is flat. That is because the phase response of a network directly affects the relative time that various frequencies of a complex modulated signal take to pass through the network. Consider the signal in Figure 1.4.

Figure 1.4 Modulated signal through a network showing distortion due to only phase shift: normal (upper), shifted (lower).

For this network, the phase of S₂₁ defines how much shift occurs for each frequency element in the modulated signal. Even though the amplitude response is the same in both Figure 1.4a,b, the phase response is different, and the envelope of the resulting output is changed. In general, there is some delay from the input to the output of a network, and the important definition that is most commonly used is the group delay of the network, defined as

      (1.45)

      While easily defined, the group delay response may be difficult to measure and/or interpret. This is because measurement instruments record discrete values for phase, and the group delay is a derivative of the phase response. Using discrete differentiation can generate numerical difficulties; Chapter 5 shows some of the difficulties encountered in practice when measuring group delay, as well as some solutions to these difficulties.

      For most complex signals, the ideal goal for phase response of a network is that of a linear phase response. Deviation from linear phase is a figure of merit for the phase flatness of a network, and this is closely related another figure of merit, group delay flatness. Thus, the ideal network has a flat group delay, meaning a linear phase response. However, many complex communications systems employ equalization to remove some of the phase response effects. Often, this equalization can account for first‐ or second‐order deviations in the phase; thus, another figure of merit is deviation from parabolic phase, which is effectively a measure of the quality of fit of