Читать книгу Handbook of Microwave Component Measurements - Joel P. Dunsmore - Страница 21

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 the phase response to a second order polynomial. These measurements are discussed further in Chapter 5.