Active Components
How to determine an effective damping
factor for a third-order PLL
A clever means for calculating (ζe) without using circuit simulations is presented.
The advantage is that a relatively dense plot (55 points) of (ζe) vs. phase margin
can be produced in a matter of seconds.
By Ken Gentile
U
(s + α )
(1)
s2
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:
H CL ( s ) =
H OL ( s )
1+ H OL ( s )
=
K (s + α )
s 2 + Ks + K α
=
 ω 
K  s+ n 
 2ζ 
s 2 + 2ζω n s + ω n2
(2)
Note the introduction of two new variables: ζ (the damping factor) and ωn (the natural frequency of the loop). Both are expressed
in terms of K and α, where
and
. 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:
K (s + α )
H OL ( s ) = s 2 (s + β )
(3)
In equation 3, α defines the zero and β the pole of the open-loop
response. This leads to a closed-loop response of:
H CL ( s ) =
H OL ( s )
1+ H OL ( s )
=
K (s + α )
s3 + β s 2 + Ks + K α
(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 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 (ωc), 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 (ζe) 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 (fTRAN) and a specified
maximum frequency error (fε) with respect to the final settling point
such that
(where fε < fTRAN).
32
706RFDF3.indd 32
1.4
1.2
1
0.8
0.6
�=30
�=50
�=70
0.4
0.2
0
0
0.5
1
1.5
2
Time (normalized)
2.5
3
Figure 1. Transient response of a third-order PLL.
0.6
0.4
Frequency (normalized)
H OL ( s ) = K
1.6
Frequency (normalized)
se 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 ζ.
The damping factor appears in the closed-loop response of a secondorder PLL, which may be derived from the open-loop response. In the
s-domain, the open-loop response has the form:
0.2
0
-0.2
-0.4
�=30
�=50
�=70
-0.6
-0.8
-1
0
0.5
1
1.5
2
Time (normalized)
2.5
3
Figure 2. Transient frequency error as a function of time.
Effective damping factor
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 (fTRAN) on the vertical and to 1/fc (or 2π/ωc) on the horizontal.
The most notable aspect of these curves is that φ has a direct impact
www.rfdesign.com
June 2007
6/21/2007 4:00:45 PM
�
��
��
����
����
����
��
��
��
�����������������������������������
�����������������������������������
�
��
��
��
��
��
���
���
���
���
��
��
��
��
��
��
�����������������
��
��
��
��
��
��
��
��
��
���
���
���
���
��
����
����
����
��
��
��
��
��
��
��
�����������������
��
��
��
Figure 3. Logarithmic transient frequency error as a function of time.
Figure 4. Approximating the slope of the envelope.
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 ζ 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,” ζe. In fact,
Vaucher[1] makes the argument that ζe 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.
generating the transient error data is relatively straightforward, but a
computational method for identifying the envelope in order to determine
the slope is not trivial.
My solution to the latter problem is based on the following observation. Given a particular value of φ and specified frequency error
threshold level, draw a line from the origin to the point on the frequency
error 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 each
trace envelope. The arrow-tipped lines indicate the aforementioned
approximation. For this example, an arbitrary threshold level of
-9.5 nepers was chosen. Notice that the slope of each arrowed line is
a reasonable approximation to the slope of the associated envelope.
Hence, it is reasonable that the slope of the arrowed line can be used to
approximate ζe instead 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 determination
of the actual slope of the envelope. Using this method for estimating
the slope of the envelope, the details of the ζe computational 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 required
to determine ζe. 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, α
and β coefficients in the closed-loop frequency response as given by
equation 4. The coefficients are calculated based on the methodology
given in references 2 and 3. Normally, in addition to the desired phase
margin, the open-loop bandwidth (ωc) is required to calculate the
coefficients. However, the open-loop bandwidth simply scales all of
the computations, so the analysis can be accomplished by normalizing
the bandwidth to unity (fc = 1 Hz) and scaling the bandwidth dependent
results by fc. The normalized values of K, α and β (as a function of
φ) are given by:
Computational algorithm
To my knowledge, nowhere in the published literature on this subject is there a closed-form solution for relating φ to ζe. Hence, some
method must be employed that can generate transient error data for a
given φ. Once the transient error data is available, a method for determining the slope of the logarithmic error envelope must be resolved,
as ζe is directly related to the slope of the envelope. The method for
34
706RFDF3.indd 34
K = (2π )
2
1 + sin (φ )
1 − sin (φ )
 1 − sin (φ )
α = 2π 
 cos (φ ) 
 cos (φ ) 
β = 2π 
 1 − sin (φ )
www.rfdesign.com
June 2007
6/21/2007 4:00:49 PM
���������� ����������
���������
�������
��������
��������
�
���������
����
������������
���������
����
����
���������� ����������
����
�����
�������� ��������
��������
����
��������
����������
�����������
���������
���
����������
�����������
�����
��������
�
���������
�������
�������
���������
��
��
�������
���������
�������
����
���������
�������
����
����
������
Figure 5. Computational process to determine ζe.
The previous equations show that φ is an argument to several trigonometric functions. Normally, φ is specified in degrees, but it should be
noted that most programming languages will require that φ be converted
to radian units to properly compute the trigonometric functions.
The closed-loop response, H(s), is computed per equation 4 using the above coefficients. For computational purposes, the Laplace
variable, s, is replaced by jω (or j2πf). The reason for generating the
closed-loop response is to ultimately derive the transient response,
from which follows the error response and its associated envelope (as
demonstrated in Figure 4). In order to produce an error response with
sufficient resolution over a broad range of phase margin values it is
necessary to compute the closed-loop response over a sufficiently wide
frequency range. It was found that a frequency span covering 700 times
the closed-loop bandwidth is sufficient for most cases. With the closedloop bandwidth normalized to 1 Hz this equates to 0 ≤ f ≤ 700.
The closed-loop response is transformed into the time domain by
means of an inverse FFT. The result is the impulse response of the
closed-loop response. However, an inverse FFT applied directly to H(s)
without modification will result in a complex impulse response. This is
due 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 imaginary
numbers), H(s) must be modified to include the appropriate negative
frequencies, as well. Doing so will cause the inverse FFT to produce
an impulse response that contains only real values. Furthermore, in
order to produce an impulse response with adequate time resolution
it is necessary to ensure that H(s) contains enough frequency samples
before invoking the inverse FFT. It was found that computing H(s) with
215 frequency points over the range -700 ≤ f ≤ 700 yields satisfactory
results. The result is that the time resolution for the impulse response
is 1/1400 in normalized units. In units of seconds, this equates to
1/(1400fc), where fc is the open-loop response in Hz.
The step response is computed by convolving the impulse response
with a step function. The step function is nothing more than a vector
of ones that is the same length as the impulse response. This yields
a step response that is normalized to unity, but it scales linearly with
any arbitrary transient step size.
The error response is calculated by subtracting 1 from normalized
step response.
The logarithmic error response is calculated from the error response.
36
706RFDF3.indd 36
However, since the error response contains positive and negative values,
the logarithmic error response is computed using the absolute value
of the error response. The result is a data set similar to that shown
in Figure 3, but containing only a single trace associated with the
specified value of φ.
Finally, the normalized logarithmic error response is analyzed to
find the point at which the data remains below the threshold level.
The normalized logarithmic threshold level is specified in nepers. It
represents the ratio of the absolute maximum allowable frequency
error to the magnitude of the initial frequency step transient. For
example, if the initial frequency step transient is 7 kHz and maximum
allowable frequency error for declaring frequency lock is 5 Hz, then
the threshold level is calculated as ln(5/7000) = -7.24 nepers. Given
the normalized logarithmic threshold level (NTHRESH) in nepers, it is
possible to find the normalized time point (tTHRESH) at which the data
crosses permanently through NTHRESH. Then, the slope of one of the
arrow-tipped lines in Figure 4 is calculated as:
Since the effective damping factor (ζe) 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) required
for a specified lock time, then the desired lock time (tLOCK) must be
provided. With tLOCK specified in units of seconds, the minimum loop
bandwidth (in hertz) is expressed as:
Effective damping factor as a function of phase
margin
With the methodology outlined above it is possible to compute ζe
over a range of φ values for a specified threshold level. This provides a
means to build a plot of ζe vs. φ that can serve as a tool for identifying
the effective damping factor associated with a particular phase margin
www.rfdesign.com
June 2007
6/21/2007 4:00:52 PM
���
�
���
�
�
���
�������������������
������������������������
���
���
���
���
�
�
�
���
���
���
�
���
���
���
���
���
��
�
���
����������������������
���
Figure 6. Effective damping factor vs. phase margin.
��
��
���
���
��
�����
�
�
�
�
NTHRESH = -10 nepers
�
���
tLOCK = 100µs
���
���
���
����������������������
Figure 8. Minimum open-loop bandwidth vs. phase margin.
The results given in reference 1 demonstrate that an “effective”
damping factor (ζe) can be determined for a third-order PLL that is
somewhat analogous to the commonly used damping factor parameter
(ζ) in second-order PLLs. In reference 1, circuit simulations were used
to generate the data for the time domain transient waveforms required
to determine ζe. For a given phase margin (φ), multiple simulations
were executed with various combinations of the transient step size
and closed-loop bandwidth (ωc) and the results averaged to arrive at a
mean value of ζe for the specified φ. The technique described builds on
the work given in reference 1 by eliminating the need to run multiple
circuit simulations. Instead, the time domain waveforms are generated
from the closed-loop transfer function after determining the necessary
coefficients as described in references 2 and 3. This technique allows
the analysis to be normalized to the closed-loop bandwidth and the
transient frequency step size, which eliminates the need for multiple
simulations and averaging. The caveat is the introduction of the small
error associated with approximating the slope of the logarithmic error
envelope rather than using the actual slope. However, the relatively
small error introduced by this approximation is worth the dramatic
reduction in processing time compared to the methodology used
in reference 1. In fact, the procedure outlined above that produced
Figure 6 through Figure 8 was executed in less than 10 seconds using
a PC with a 2.4 GHz dual-core processor. RFD
706RFDF3.indd 38
���
��
∆φ = 0.5º
38
���
��
φmax = 80º
Conclusion
���
����������������������
Figure 7. Settling time vs. phase margin.
�������
value (based on a specific threshold level). Since the above procedure
yields tTHRESH and fc_min, then it is a simple matter to also generate plots
for tTHRESH vs. φ and fc_min vs. φ.
Figure 6 through Figure 8 are plots that were generated using
MATLAB to execute the procedure outlined above. The dashed trace
indicates the raw data generated by the procedure. The solid trace
is data after smoothing. The ripple that appears in the raw data can
be attributed to the approximation of the slope of the envelope (the
arrowed lines in Figure 4) rather than the true slope (the dashed lines
in Figure 4). The results agree reasonably well with those presented
in reference 1. The slight deviation in the values shown in the plots
here with respect to those shown in reference 1 can be attributed to
the same slope approximation error mentioned above.
The plots here were generated with the following parameters:
φmin = 15º
���
References
1. Vaucher, C. S., “An Adaptive PLL Tuning System Architecture
Combining High Spectral Purity and Fast Settling Time,” IEEE Journal
of Solid-State Circuits, vol. 35, No. 4, April 2000.
2. Hawkins, D. W., “Digital Phase-Locked Loop (PLL)-Based
Frequency Synthesizers: Theory and Analysis,” June 18, 1999, World
Wide Web.
3. Keese, W. O., “An Analysis and Performance Evaluation of a
Passive Filter Design Technique for Charge Pump Phase-Locked
Loops,” National Semiconductor Application Note 1001, May 1996.
4. Wolaver, D. H., Phase-Locked Loop Circuit Design, Prentice
Hall, 1991.
ABOUT THE AUTHOR
Ken Gentile is a system design engineer for the Clock and Signal
Synthesis Products Group at Analog Devices, Greensboro, NC.
His specialties are the application of digital signal-processing techniques in communications systems and analog filter
design. He holds a B.S.E.E. degree from North Carolina State
University.
www.rfdesign.com
June 2007
6/21/2007 4:00:55 PM