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32 www.rfdesign.com June 2007

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How to determine an effective dampingfactor for a third-order PLLA clever means for calculating (

e) without using circuit simulations is presented.

The advantage is that a relatively dense plot (55 points) of (e) vs. phase margin

can be produced in a matter of seconds.

By Ken Gentile

Use of the damping factor () parameter as a gauge of the transientresponse of a second-order feedback loop is common in control

theory. As such, it is a common practice to define the transient char-acteristics 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:( )

2( )s

OL sH s K

+= (1)

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

( )2 2 2

2( )

1 ( ) 2( )

n

OL

OL n n

K sK sH s

CL H s s Ks K s sH s

+ +

+ + + + += = = (2)

Note the introduction of two new variables: (the damping fac-tor) and

n(the natural frequency of the loop). Both are expressed

in terms of K and , where and . The value ofcorrelates 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:

( )( )2( )

K s

OL s sH s

+

+= (3)

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

( )3 2

( )

1 ( )( ) OL

OL

K sH s

CL H s s s Ks KH s

++ + + +

= = (4)

Because the denominator takes the form of a cubic polynomial in s,the concept of a damping factor no longer makes sense. This is becausea cubic has three factors, which, in general, would require three newvariables defined in terms of K, and . With three variables defin-ing the loop response, the transient behavior would be determined bythe interaction of at least two of the variables. This precludes the useof a single variable as a gauge for the transient behavior of the loop.Hence, third-order PLLs are defined in terms of phase margin () andopen-loop bandwidth (

c), which are related to and 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 an ef-fective damping factor (

e) can be obtained if one specifies a maximum

amount of fractional settling error, . That is, can be expressed interms of a specified transient frequency step size (f

TRAN) and a specified

maximum frequency error (f) with respect to the final settling pointsuch that (where f < fTRAN).

Effective damping factorFigure 1 shows the transient behavior of a third-order PLL for three

different values of. The plot is normalized to the frequency transientstep size (f

TRAN) on the vertical and to 1/f

c(or 2/

c) on the horizontal.

The most notable aspect of these curves is that has a direct impact

=30

=50

=70

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

00 0.5 1 1.5 2 2.5 3

Time (normalized)

Frequency(normalized

)

Figure 1. Transient response of a third-order PLL.

=30

=50

=70

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

-10 0.5 1 1.5 2 2.5 3

Time (normalized)

Frequency(normalized)

Figure 2. Transient frequency error as a function of time.

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on the overshoot and settling characteristics of the loop. So it would

seem reasonable that some corollary could be drawn between in athird-order system and in a second-order system. This was shownto be true according to Vaucher[1].

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

Although helpful in visualizing the transient error, Figure 2 doesnot provide much insight into an analytical solution for relating to .However, if the transient error is plotted on a log-scale, an interestingobservation 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 muchwider dynamic range of frequency error. Also, dashed lines have beenadded that indicate the slope of the envelope of each of the traces. Thestraight-line nature of the trace envelopes is an important observation.Notice that the slope of the envelope provides a linear relationshipbetween the logarithmic frequency error and time. That is, given a valueof (which defines a particular trace) and some specified maximumacceptable relative error threshold (in nepers), we see that the timerequired to reach the threshold level may be derived from the slopeof the trace envelope. It appears that the slope of the envelope is theconnection between and an effective damping factor,

e. In fact,

Vaucher[1] makes the argument that ecan be defined as the inverse of

the slope of the envelope, where the envelope is defined by observingthe normalized logarithmic frequency error as a function of time.

It is also interesting to observe that as increases from 30 to 50,the slope of the envelope becomes steeper. However, as

increases

from 50 to 70, the slope becomes shallower. This would imply thatthere may be some optimal value of that yields the quickest time tosettle to a given error threshold level. In fact, this is shown to be trueand is demonstrated at the conclusion of this article.

Computational algorithmTo my knowledge, nowhere in the published literature on this sub-

ject is there a closed-form solution for relating to e. Hence, some

method must be employed that can generate transient error data for agiven . Once the transient error data is available, a method for deter-mining the slope of the logarithmic error envelope must be resolved,as

eis directly related to the slope of the envelope. The method for

generating the transient error data is relatively straightforward, but a

computational method for identifying the envelope in order to determinethe slope is not trivial.

My solution to the latter problem is based on the following obser-vation. Given a particular value of and specified frequency errorthreshold level, draw a line from the origin to the point on the frequencyerror trace where the frequency error first falls below the threshold level.This line is a fairly good approximation of the slope of the envelope.This is shown in Figure 4.

In keeping with Figure 3, a dashed line indicates the slope of eachtrace envelope. The arrow-tipped lines indicate the aforementionedapproximation. For this example, an arbitrary threshold level of-9.5 nepers was chosen. Notice that the slope of each arrowed line isa reasonable approximation to the slope of the associated envelope.Hence, it is reasonable that the slope of the arrowed line can be used toapproximate

einstead of the actual slope of the envelope. The reason

for using this approximation is that determination of the slope of the

arrowed line is a much more tractable problem than the determinationof the actual slope of the envelope. Using this method for estimatingthe slope of the envelope, the details of the

ecomputational process

may be addressed as shown in Figure 5.The input parameters are phase margin, normalized logarithmic

threshold level and lock time. Only the first two parameters are requiredto determine

e. The lock time is only required if a calculation of the

minimum necessary loop bandwidth is desired.The phase margin is used as the parameter to determine the K,

and coefficients in the closed-loop frequency response as given byequation 4. The coefficients are calculated based on the methodologygiven in references 2 and 3. Normally, in addition to the desired phasemargin, the open-loop bandwidth (

c) is required to calculate the

coefficients. However, the open-loop bandwidth simply scales all ofthe computations, so the analysis can be accomplished by normalizingthe bandwidth to unity (f

c= 1 Hz) and scaling the bandwidth dependent

results by fc. The normalized values of K, and (as a function of

) are given by:

( ) ( )( )2 1 sin

K 21 sin

+ =

( )( )

1 sin2

cos

=

( )( )

cos2

1 sin

=

Figure 3. Logarithmic transient frequency error as a function of time. Figure 4. Approximating the slope of the envelope.

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The previous equations show that is an argument to several trigo-nometric functions. Normally, is specified in degrees, but it should benoted that most programming languages will require that be convertedto radian units to properly compute the trigonometric functions.

The closed-loop response, H(s), is computed per equation 4 us-ing the above coefficients. For computational purposes, the Laplacevariable, s, is replaced by j(or j2f). The reason for generating theclosed-loop response is to ultimately derive the transient response,from which follows the error response and its associated envelope (asdemonstrated in Figure 4). In order to produce an error response withsufficient resolution over a broad range of phase margin values it is

necessary to compute the closed-loop response over a sufficiently widefrequency range. It was found that a frequency span covering 700 timesthe closed-loop bandwidth is sufficient for most cases. With the closed-loop bandwidth normalized to 1 Hz this equates to 0 f 700.

The closed-loop response is transformed into the time domain bymeans of an inverse FFT. The result is the impulse response of theclosed-loop response. However, an inverse FFT applied directly to H(s)without modification will result in a complex impulse response. This isdue to the fact that H(s) is usually expressed for positive frequencies.Since we know that the impulse response must be real (i.e., no imaginarynumbers), H(s) must be modified to include the appropriate negativefrequencies, as well. Doing so will cause the inverse FFT to producean impulse response that contains only real values. Furthermore, inorder to produce an impulse response with adequate time resolutionit is necessary to ensure that H(s) contains enough frequency samplesbefore invoking the inverse FFT. It was found that computing H(s) with215 frequency points over the range -700 f 700 yields satisfactoryresults. The result is that the time resolution for the impulse responseis 1/1400 in normalized units. In units of seconds, this equates to1/(1400f

c), where f

cis the open-loop response in Hz.

The step response is computed by convolving the impulse responsewith a step function. The step function is nothing more than a vectorof ones that is the same length as the impulse response. This yieldsa step response that is normalized to unity, but it scales linearly withany arbitrary transient step size.

The error response is calculated by subtracting 1 from normalizedstep response.

The logarithmic error response is calculated from the error response.

However, since the error response contains positive and negative values,the logarithmic error response is computed using the absolute valueof the error response. The result is a data set similar to that shownin Figure 3, but containing only a single trace associated with thespecified value of.

Finally, the normalized logarithmic error response is analyzed tofind the point at which the data remains below the threshold level.The normalized logarithmic threshold level is specified in nepers. Itrepresents the ratio of the absolute maximum allowable frequencyerror to the magnitude of the initial frequency step transient. Forexample, if the initial frequency step transient is 7 kHz and maximum

allowable frequency error for declaring frequency lock is 5 Hz, thenthe threshold level is calculated as ln(5/7000) = -7.24 nepers. Giventhe normalized logarithmic threshold level (N

THRESH) in nepers, it is

possible to find the normalized time point (tTHRESH

) at which the datacrosses permanently through N

THRESH. Then, the slope of one of the

arrow-tipped lines in Figure 4 is calculated as:

Since the effective damping factor (e) may be approximated as the

inverse of the slope of the arrow-tipped line, then:

If one wishes to know the minimum loop bandwidth (fc_min

) requiredfor a specified lock time, then the desired lock time (t

LOCK) must be

provided. With tLOCK

specified in units of seconds, the minimum loopbandwidth (in hertz) is expressed as:

Effective damping factor as a function of phasemargin

With the methodology outlined above it is possible to compute e

over a range of values for a specified threshold level. This provides ameans to build a plot of

evs. that can serve as a tool for identifying

the effective damping factor associated with a particular phase margin

Figure 5. Computational process to determine e.

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value (based on a specific threshold level). Since the above procedureyields t

THRESHand f

c_min, then it is a simple matter to also generate plots

for tTHRESH vs. and fc_min vs. .Figure 6 through Figure 8 are plots that were generated usingMATLAB to execute the procedure outlined above. The dashed traceindicates the raw data generated by the procedure. The solid traceis data after smoothing. The ripple that appears in the raw data canbe attributed to the approximation of the slope of the envelope (thearrowed lines in Figure 4) rather than the true slope (the dashed linesin Figure 4). The results agree reasonably well with those presentedin reference 1. The slight deviation in the values shown in the plotshere with respect to those shown in reference 1 can be attributed tothe same slope approximation error mentioned above.

The plots here were generated with the following parameters:

min= 15

max

= 80

= 0.5

NTHRESH

= -10 nepers

tLOCK

= 100s

ConclusionThe results given in reference 1 demonstrate that an effective

damping factor (e) can be determined for a third-order PLL that is

somewhat analogous to the commonly used damping factor parameter() in second-order PLLs. In reference 1, circuit simulations were usedto generate the data for the time domain transient waveforms requiredto determine

e. For a given phase margin (), multiple simulations

were executed with various combinations of the transient step sizeand closed-loop bandwidth (

c) and the results averaged to arrive at a

mean value ofefor the specified . The technique described builds on

the work given in reference 1 by eliminating the need to run multiplecircuit simulations. Instead, the time domain waveforms are generatedfrom the closed-loop transfer function after determining the necessarycoefficients as described in references 2 and 3. This technique allowsthe analysis to be normalized to the closed-loop bandwidth and thetransient frequency step size, which eliminates the need for multiplesimulations and averaging. The caveat is the introduction of the smallerror associated with approximating the slope of the logarithmic errorenvelope rather than using the actual slope. However, the relativelysmall error introduced by this approximation is worth the dramaticreduction in processing time compared to the methodology usedin reference 1. In fact, the procedure outlined above that producedFigure 6 through Figure 8 was executed in less than 10 seconds usinga PC with a 2.4 GHz dual-core processor.RFD

References1. Vaucher, C. S., An Adaptive PLL Tuning System Architecture

Combining High Spectral Purity and Fast Settling Time, IEEE Journalof Solid-State Circuits, vol. 35, No. 4, April 2000.

2. Hawkins, D. W., Digital Phase-Locked Loop (PLL)-BasedFrequency Synthesizers: Theory and Analysis, June 18, 1999, WorldWide Web.

3. Keese, W. O., An Analysis and Performance Evaluation of aPassive Filter Design Technique for Charge Pump Phase-LockedLoops, National Semiconductor Application Note 1001, May 1996.

4. Wolaver, D. H., Phase-Locked Loop Circuit Design, PrenticeHall, 1991.

Figure 6. Effective damping factor vs. phase margin.

Figure 7. Settling time vs. phase margin.

Figure 8. Minimum open-loop bandwidth vs. phase margin.

ABOUT THE AUTHOR

Ken Gentile is a system design engineer for the Clock and SignalSynthesis Products Group at Analog Devices, Greensboro, NC.His specialties are the application of digital signal-process-ing techniques in communications systems and analog filterdesign. He holds a B.S.E.E. degree from North Carolina StateUniversity.

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