rel="nofollow" href="#u8524e5e7-251a-5b88-928b-380600ace97f">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 the phase response to a second order polynomial. These measurements are discussed further in Chapter 5.
1.4 Power Parameters
1.4.1 Incident and Reflected Power
Just as there are a variety of S‐parameters, which are derived from the fundamental parameters of incident and reflected waves a and b, so too are there many power parameters that can be identified with the same waves. As inferred earlier, the principal power parameters are incident and reflected, or forward and reverse, powers at each port, which for Z0 real, are defined as
(1.46)
The proper interpretation of these parameters is that incident and reflected power is the power that would be delivered to a nonreflecting (Z0) load. If one were to put an ideal Z0 directional‐coupler in line with the signal, it would sample or couple the incident signal (if the coupler were set to couple the forward power) or the reflected signal (if the coupler were set to couple the reverse power). In simulations, ideal directional‐couplers are often used in just such a manner.
1.4.2 Available Power
The maximum power that can delivered from a generator is called the available power, or PAvailable, and can be defined as the power delivered from a ZS
where ΓS is computed as in Eq. (1.24) as
(1.48)
This maximum power is delivered to the load when the load impedance is the conjugate of the source impedance,
1.4.3 Delivered Power
The power that is absorbed by an arbitrary load is called the delivered power, and it is computed directly from the difference between the incident and reflected power.
(1.49)
For most cases, this is the power parameter that is of greatest interest. In the case of a transmitter, it represents the power that is delivered to the antenna, for example, which in turn is the power radiated less the resistive loss of the antenna.
1.4.4 Power Available from a Network
A special case of available power is the power available from the output of a network, when the network is connected an arbitrary source. In this case, the available power is only a function of the network and the source impedance and is not a function of the load impedance. It represents the maximum power that could be delivered to a load under the condition that the load impedance was ideally matched and can be found by noting that the available output power is similar to Eq. (1.47) but with the source reflection coefficient replaced by the output reflection coefficient of the network Γ2 from Eq. (1.30) such that
