Use of the damping factor (ζ) parameter as a gauge of the transient response of a second-order feedback loop is common in control theory. As such, it is a common practice to define the transient characteristics of a second-order phase-locked loop (PLL) in terms of . The damping factor appears in the closed-loop response of a second-order PLL, which may be derived from the open-loop response. In the s-domain, the open-loop response has the form:

Where s is the complex frequency variable associated with the Laplace transform, α defines the zero of the open-loop response and K is the open-loop gain. This leads to a closed-loop response of the form:

Note the introduction of two new variables: ζ (the damping factor) and ω_n (the natural frequency of the loop). Both are expressed in terms of K and α, where ζ = √K/4α and ω_n = √Kα. The value of ζ correlates directly to the settling characteristics (transient response) of a second-order PLL, which is what makes it an attractive loop control parameter.

A third-order PLL, on the other hand, has an open-loop response, which has the form:

In equation 3, α defines the zero and β the pole of the open-loop response. This leads to a closed-loop response of:

Because the denominator takes the form of a cubic polynomial in s, the concept of a damping factor no longer makes sense. This is because a cubic has three factors, which, in general, would require three new variables defined in terms of K, α and β. With three variables defining the loop response, the transient behavior would be determined by the interaction of at least two of the variables. This precludes the use of a single variable as a gauge for the transient behavior of the loop. Hence, third-order PLLs are defined in terms of phase margin (φ) and open-loop bandwidth (ω_c), which are related to α and β as given in equations 2 and 3.

Even though third-order loops do not lend themselves to a damping factor parameter, Vaucher^[1] showed that for a given value of φ an effective damping factor (ζ_e) can be obtained if one specifies a maximum amount of fractional settling error, ε. That is, ε can be expressed in terms of a specified transient frequency step size (f_TRAN) and a specified maximum frequency error (f_ε) with respect to the final settling point such that ε = f_ε/f_TRAN (where f_ε < f_TRAN).

Figure 1 shows the transient behavior of a third-order PLL for three different values of φ. The plot is normalized to the frequency transient step size (f_TRAN) on the vertical and to 1/f_c (or 2π/ω_c) on the horizontal. The most notable aspect of these curves is that φ has a direct impact on the overshoot and settling characteristics of the loop. So it would seem reasonable that some corollary could be drawn between φ in a third-order system and in a second-order system. This was shown to be true according to Vaucher^[1].

It is generally understood that ζ is a parameter related to the time required for a second-order PLL to settle to some acceptable level of frequency error following a transient frequency step. Since our goal is to relate φ to an “effective” ζ, it makes sense to view the transient step response in terms of frequency error relative to the final steady-state value. This is shown in Figure 2 for the same three values of φ. Note that the steady-state value corresponds to 0 on the vertical scale and the traces now display deviation from steady state as a function of time.

Although helpful in visualizing the transient error, Figure 2 does not provide much insight into an analytical solution for relating ζ to φ. However, if the transient error is plotted on a log-scale, an interesting observation can be made. This is shown in Figure 3.

Note that the horizontal axis has been extended, because plotting the transient error logarithmically makes it easier to view a much wider dynamic range of frequency error. Also, dashed lines have been added that indicate the slope of the envelope of each of the traces. The straight-line nature of the trace envelopes is an important observation. Notice that the slope of the envelope provides a linear relationship between the logarithmic frequency error and time. That is, given a value of φ (which defines a particular trace) and some specified maximum acceptable relative error threshold (in nepers), we see that the time required to reach the threshold level may be derived from the slope of the trace envelope. It appears that the slope of the envelope is the connection between φ and an “effective damping factor,” ζ_e. In fact, Vaucher^[1] makes the argument that ζ_e can be defined as the inverse of the slope of the envelope, where the envelope is defined by observing the normalized logarithmic frequency error as a function of time.

It is also interesting to observe that as φ increases from 30° to 50°, the slope of the envelope becomes steeper. However, as φ increases from 50° to 70°, the slope becomes shallower. This would imply that there may be some optimal value of φ that yields the quickest time to settle to a given error threshold level. In fact, this is shown to be true and is demonstrated at the conclusion of this article.

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