Accurate Behavioral Modeling Approach
for PLL Designs with Supply Noise Effects
Chin-Cheng Kuo and Chien-Nan Jimmy Liu
Department of Electrical Engineering
National Central University, Taiwan, ROC
92521014@cc.ncu.edu.tw; jimmy@ee.ncu.edu.tw
addition, some detailed circuit properties, such as timing
information and parasitic effects, are hard to be modeled in
the top-down approach due to the lack of layout
information. Therefore, for IP-based SOC designs, bottomup extraction approaches [4, 5] that build the behavioral
models from post-layout simulation results may be the
better way to obtain more accurate models for existing
designs.
In our previous work [5], an efficient approach is
presented to generate the behavioral models of PhaseLocked Loop (PLL) designs more quickly. The main idea
is to use a special “characterization mode” to get required
circuit parameters. In the characterization mode, only one
input pattern is enough to get circuit properties with
parasitic effects and intrinsic noises. No time-consuming
correlation analysis is required to build this accurate model.
As shown in the experimental results, the behavioral
models built by this approach can have accurate results,
especially on those non-ideal effects such as output jitters.
In SoC designs, the large portion of digital circuits
may require large current to drive. This large current may
incur many problems, such as the well-known IR-drop
problem, that change the value of supply voltage. This kind
of voltage fluctuation may have large impacts on the
performance and reliability of designs, especially for
analog circuits. Therefore, after the mixed-signal designs
are integrated, supply voltage variation must be considered
to evaluate the system performance accurately. However,
those bottom-up approaches only deal with ideal
environments without any variation. In order to have more
realistic simulation results, we do need another approach to
model the performance changes under non-ideal
environments.
Actually, without detailed low-level simulation of the
whole design, the possible variation on supply voltage is
hard to be predicted. If we are going to estimate the
responses under such voltage fluctuation at behavioral level,
two primary approaches are often used. One is making an
assumption that supply voltage variation is a random
distribution and adding a Gaussian or Uniform variation
generator into the behavioral models [6]. This is a simpler
way to have some responses for observation. However,
they may not the real cases while integrated into the system
because the drawn current of the digital part may not fit the
Abstract
Using behavioral models to perform system
simulation at behavioral level is currently a popular
solution to verify mixed-signal systems. However, most of
existing approaches only deal with ideal environments or
make unrealistic assumptions, which cannot accurately
evaluate the performance under supply voltage variation. In
this paper, an efficient modeling approach with the effects
of supply voltage variation is presented for PLL circuits to
build accurate behavioral models in a short time using
bottom-up extraction. Only three post-layout simulations
are enough to generate accurate behavioral models under
different supply voltages. The experimental results have
shown that this approach can really have accurate
responses under different supply voltages without timeconsuming correlation analysis.
1. Introduction
In recent years, analog mixed-signal designs are very
popular in ASIC applications. How to integrate digital and
analog designs correctly becomes a serious problem. One
of the popular ways to solve above issues is building the
behavioral models for both digital and analog circuits and
performing system simulation at behavioral level. Using
those behavioral models, we can finish whole chip
simulation in a very short time such that we can verify the
integrated system earlier before down to layout level and
reduce the design iterations.
In the literature, researches on building behavioral
models of analog circuits are very popular [1, 2, 3].
According to the specifications, designers can use
Simulink/Matlab or analog HDL to describe the
mathematic formulas of analog circuits and simulate the
possible results before implementation. Typically, such
kind of top-down modeling approach is helpful for circuit
designers to estimate the final results much faster when
they are building their new designs.
In current SOC era, design reuse or IP-based design
methodology is a popular solution to handle complex
system. Such kind of verified blocks, especially for analog
IPs, often appear as layout form or post-layout netlists,
which is hard to understand without designers’ help. In
____________________
This work was supported in part by R.O.C. National Science Council under Grant NSC93-2215-E008-031 and Ministry of Economic Affairs under Grant MOEA94-EC-17-A-01-S1-037.
0-7803-9352-X/05/$20.00 © 2005 IEEE.
48
For PLL designs, the proposed characterization mode
is to break the PLL loop without separating it into
independent blocks as shown in Figure 2. The broken
connection is also very helpful for us to send special
patterns to trigger the PLL into different situations.
Therefore, we only need one input pattern to trigger the
design and extract all the critical characteristic parameters.
In the following subsections, the special input pattern used
to trigger the PLL is presented. The relative output
responses, the extracted characteristic parameters, and their
meanings to our models will also be explained in detail.
Gaussian distribution very well. Therefore, another
approach takes the real supply voltage value as another
input of the behavioral model and computes the real
responses at this supply voltage immediately [7]. However,
building such kind of behavioral models often requires
statistical analysis, such as Monte Carlo (MC) analysis [8],
to fit a curve of real responses under different supply
voltages. Since a lot of simulation results are required in
the statistical analysis, the construction time of such
behavioral models will become unacceptable if post-layout
simulation is used to obtain accurate results.
In this paper, an efficient modeling approach with the
effects of supply voltage variation is presented for PLL
designs to build accurate behavioral models in a short time
using bottom-up extraction. Because all the parameters in
the behavioral models proposed in our previous work [5]
are obtained from voltage-domain measurement, their
changes under supply voltage variation are almost
proportional to the value changes of supply voltages in our
observation. Therefore, using our modeling approach, we
only have to run three post-layout simulations to obtain
three behavioral models when the supply voltage has no
change, positive change, and negative change. Then, given
any supply voltage value, we can determine the parameter
values of the behavioral model using interpolation or
extrapolation from the parameter values of the three models.
And the real responses under this situation can be obtained
quickly by simulating the generated behavioral model as
shown in Figure 1. According to the experimental results
presented in Section 4, this approach can really have
accurate responses under different supply voltages without
tedious post-layout simulations.
Figure 2. Characterization Mode of PLL
Please note that no exact current values and device
sizes are required in the following measurement process.
Since voltage-domain behavioral models are used, all
parameters can be measured from the simulation results in
the special characterization mode. Moreover, because we
do not separate the netlist into sub-blocks in the simulation,
tedious layout tracing step can be avoided and the
interactions between blocks can be automatically
considered. We believe that this would be a more suitable
approach for existing IPs and flattened netlists.
2.1 Modeling Process
The proposed special extracting pattern is illustrated
in Figure 3. The positive edge of Fref will create positive
phase error such that the source current of CP can charge
Vctrl in stage 1. If we give enough time, we can charge Vctrl
until voltage high to observe the entire response. Almost all
characteristic parameters can be extracted in stage 1. The
other positive edge of Fb will let the PFD operate in the
reset state (stage 2). The reset time of PFD and the current
mismatch can be observed from the output response. In the
following subsections, we will explain how to get the
parameters of each block in these two stages.
Figure 1. Our Ideas
2. Bottom-Up Extraction Approach for PLL
In this section, we will briefly introduce the bottom-up
extraction flow proposed in our previous work [5] to
generate an accurate PLL behavioral model. Actually, in
the characterization process, the design under extraction
does not have to operate in the same manner as in real
system. If we can make the design operate in a special
mode such that the required parameters can be obtained
faster, which is called the “characterization mode”, it can
save much simulation time for building the behavior model.
Figure 3. Input Pattern and PFD Response
49
2.1.1 PFD & Divider
Figure 3 also shows the output responses of PFD (Up
and Dn). The parameters of delay time, reset time and
output slew-rate can be easily extracted from the output
waveforms. Those delay parameters are the primary
sources of the non-ideal effects of PFD, such as dead zone.
Similarly, the timing information of the frequency divider
block can also be observed if we measure the delay time
between Vout and the output response of divider.
V s lo p e
2.1.2 CP & LF
V dc
Figure 5. Waveform of Vctrl in Stage 1
Figure 4. CP and LF
Z LF
V c tr l
2
s R C 1C
I CP Z LF
2
s (C1 C
2
)
s(
sR C 1
E
1 E
E
E
' V c tr l
C1
'T
'I R
' V c trl
(3)
(4)
)
Mis
1
Vslope e
(5)
Vslope
I
sICP R CP
C1
s 2 R C1
1 E
)
s(
I
sICP R CP
C1
|
s
I CP R V dc
'I
Figure 6. Waveform of Vctrl with MOS Capacitances
(1)
C1
V s lo p e
About the parameter Vdc, it can be viewed as the
voltage variation of Vctrl in Equation 4. Therefore, we can
easily obtain it from the starting value of Vctrl, as shown in
Figure 5, because it is the voltage drop when the capacitors
are not charged yet.
The other non-ideal effect of CP, current mismatch,
can charge or discharge Vctrl a little bit even when the PLL
is locked. Therefore, the operation in stage 2 can be used to
obtain this mismatch information. We define another
parameter e, which is the slope of Vctrl waveform in the
lock mode as shown in Figure 7, to represent the current
mismatch. Using this parameter, we define the mismatch
parameter Mis in Equation 5 to modify the output behavior
of CP and LF in the reset situation so that the current
mismatch effects can be included in our model.
1
sR
C1
In typical designs, capacitances will be implemented
by transistors. As shown in Figure 6, there are two different
capacitance values with different responses on Vctrl when
the transistor is operated in the triode or saturation regions.
This effect can be easily handled in our model if we
measure two different values Vslope1 and Vslope2 for the
different regions. If equivalent capacitance components are
used, we can just set these two values to be the same one.
Figure 4 shows the common CP structure with a
second-order low pass filter. Because the exact values of
internal currents are hard to be observed, we use voltagedomain behavioral models that combine CP and LP
together and measure the voltage changes instead of the
actual current values. This approach allows us to consider
the parasitic effects automatically without the actual values
of equivalent RC components or pole/zero positions. The
reasons will be explained in the following formulas.
In typical designs, the capacitance C1 is often much
larger than C2 to obtain a reasonable phase margin. We
define a parameter ȕ to be C1/C2 and assume that ȕ is a
user-given large enough number. Then, we can get the
impedance of the low pass filter (ZLF) as shown in Equation
1. Using the result of Equation 1, we can have the transfer
function of Icp and Vctrl as shown in Equation 2 under the
assumption that ȕ is large enough to be ignored. In this
equation, parameter Vdc is defined as IcpR and parameter
Vslope is defined as Icp/C1. In other words, we change the
transfer function from the current form to the voltage form.
sR C1 1
I CP
(2)
E
ICP R
ICP
sC1
Vdc
V slo p e
s
In the stage 1 of the proposed input pattern, the “Up”
signal of PFD will remain logic high for a long time so that
the source current charges Vctrl up to the Vdd gradually.
According to equation 3, parameter Vslope is the rising slope
of Vctrl, as shown in Figure 5.
Figure 7. Vctrl Variations in Stage 2
2.1.3 VCO
The critical concerns of VCO circuits are the gain
(KVCO), the range of input control voltage and the output
50
frequency. In the previous approach [4], curve fitting is
used to model the behavior of VCO accurately. However,
the extracted data must be large enough to have accurate
coefficients of the fitting equations, which may increase the
required simulation time for building the VCO model.
In fact, rebuilding the completed VCO response is not
necessary because designers often prefer to operate VCO in
the linear region as possible as they can. For this reason, a
simple linear VCO model is accurate enough in the normal
situation of PLL. According to the related study [7], linear
VCO model can predict more than 90% of real VCO
characteristics. Therefore, in our approach, we adopt linear
VCO model to avoid long simulation time in traditional
curve fitting approach.
In the stage 1 of the proposed input pattern, the output
current of CP will charge Vctrl up to the Vdd gradually.
During the charging process, we can measure any two
voltage values on Vctrl (V1 and V2) and measured the
corresponding output frequencies (F1 and F2) to calculate
the gain of VCO (KVCO), as shown in Figure 8. Since the
operational range of VCO (Vmin and Vmax) is often clearly
specified in the data sheet, we can build up the linear VCO
model after the KVCO is calculated. If these two values are
not specified, we can measure the frequencies at Vctrl = 0
and Vctrl = Vdd to derive the Vmin and Vmax of this VCO
according to the relationship of the measured KVCO.
Figure 9. Supply Voltage Variation vs. Internal Parameters
Therefore, we use a Gaussian distribution as stimuli to
do 100-run MC analysis and obtain the corresponding
distributions of several important parameters of our
behavioral models. Their 1ı values are compared in Table
1 with the measurement results when the voltage change is
set as 1ı. From the results, we can see that the measured
results using our approach are very close to the timeconsuming MC simulation. Since the parameters obtained
from this assumption are accurate, our behavioral models
can still have accurate responses with those parameters.
Table 1. MC simulation (100-run) vs. Our Idea (3-run)
up_tr (s)
vco_tr (s)
div_tr (s)
up_tf (s)
vco_tf (s)
div_tf (s)
up_delay (s)
rst_delay (s)
vco_delay (s)
div_delay (s)
fmin (Hz)
fmax (Hz)
Kvco (Hz/V)
Vdc (V)
Vslope (V)
Figure 8. Relationship between Vctrl and Fout
3. Modeling the Effects of Vdd Variation
In this section, we will introduce how to use the
extraction results from only three post-layout simulations
to determine the parameter values of our behavioral model
under any given value change of supply voltage. Whatever
form of supply voltage variation can be handled in our
approach. Traditional Monte Carlo analyses can thus being
avoided in this way.
In the proposed behavioral models shown in Section 2,
all of parameters are extracted from voltage-domain
measurement such that their changes under supply voltage
variation are highly related to the variation of supply
voltage. Therefore, we make an assumption that the
distribution of those internal parameters in our behavioral
models will be similar to the distribution of supply voltage
variation, as illustrated in Figure 9. In other words, the 1ı
values of those internal parameters obtained from MC
simulation should be similar to the measurement results
when the voltage change is also 1ı.
MC (+1ı)
Measured
4.0374E-11
3.8137E-11
3.4495E-11
3.5067E-11
3.1734E-11
2.8987E-11
4.4566E-10
3.1641E-10
1.4448E-09
3.7085E-09
3.1495E+08
1.0025E+09
1.1590E+09
9.2416E-02
6.1743E+05
4.1283E-11
3.8893E-11
3.5015E-11
3.5525E-11
3.1958E-11
2.9328E-11
4.4624E-10
3.1667E-10
1.4475E-09
3.7185E-09
3.1564E+08
1.0035E+09
1.1660E+09
9.2763E-02
6.1539E+05
Error
(%)
2.25
1.98
1.51
1.31
0.70
1.18
0.13
0.08
0.19
0.27
0.22
0.10
0.61
0.38
-0.33
Because we cannot predict the distribution of supply
voltage variation accurately, we do need several postlayout simulations to have some samples on the real
responses. In our approach, we only use three simulations
to have three sets of characteristic parameters for the
behavioral models when the supply voltage has no change,
positive change, and negative change. The amount of
supply voltage variation can be chosen arbitrarily. Since
the changes of those internal parameters in our behavioral
models are almost proportional to the value changes of
supply voltage, we can determine the corresponding
parameter values of the new behavioral model given any
supply voltage changes (dvdd) using interpolation or
51
As shown in the right side of Figure 11, the
measurement results have demonstrated that the errors of
maximum overshoot (Vmax) and lock voltage (Vlock) are less
than 1% by using our behavioral model. Compared to the
results of HSPICE simulation, the lock time (settling time)
and period jitter (PJ) of PLL are also very close. However,
if we look at the simulation results of traditional behavioral
model, the obtained values are quite different to the real
values. In addition, the ǻVctrl response after locked is much
more realistic by using our behavioral model because
actual non-ideal effects have been successfully considered.
If traditional behavioral model is used, there is almost no
variation on Vctrl after locked thus cannot produce any
jitters. Those results can demonstrate that our behavioral
model has accurate results with intrinsic noises in the ideal
environment.
Then, we run the other two simulations when dvdd are
-15mV and 60mV to obtain three sets of extracting results
such that we can use interpolation or extrapolation to
determine the parameter values of our behavioral model
under any given value change of supply voltage. We use a
random variation generator to produce various voltage
variations for demonstrating our approach. In our
experiments, 60mV is chosen as the 1ı value and the ideal
case is the mean value ʻµ) for a Gaussian random generator.
The worst case sampled nearʳ ±3ı is closed to 0.1Vdd,
which is a reasonable variation value. As shown in Figure
12, which is the various settling responses on the Vctrl
waveform, our noise aware behavioral models are indeed
sensitive to Vdd variations.
extrapolation from the measured results of these three
simulations, as illustrated in Figure 10. In other words,
using Equation 6 is enough to calculate the corresponding
changes of characteristic parameters (dbehavioral). The
hundreds or thousands of MC simulations can be avoided
in this way and similar results can still be obtained.
Figure 10. Relationship of the Variations
V vdd
V behavioral
d vdd
d behavioral
d behavioral
V behavioral
V vdd d vdd
(6)
In next section, we will demonstrate our idea by a real
PLL case and use a Gaussian random generator to produce
various values of supply voltage changes. Please note that
any variation form can be handled using our approach. A
random variation generator is just to produce some
unspecified Vdd changes.
4. Experiments and Results
In order to demonstrate the accuracy of our PLL
behavioral model under supply voltage variation, we use a
PLL designed in TSMC 0.18µm CMOS process as an
experiment. The value of Fref is 50MHz and the output
frequency (Fout) is 800MHz. In our approach, we only have
to run three post-layout simulations to obtain three
behavioral models when the supply voltage has no change,
negative change, and positive change. In the experiment,
we arbitrarily choose three values as the variations of
supply voltage (dvdd), which are 0, -15mV and 60mV.
At first, we show the modeling results without supply
voltage fluctuation. The overall impacts of those non-ideal
effects often appear as the jitter form at the PLL outputs,
which is also the critical concern of designers. Moreover,
variations of Vctrl are the important factor of output jitters
such that we will focus on comparing the waveform of Vctrl,
output jitters, the lock voltage, and the lock time to the
HSPICE results in Figure 11. The traditional behavioral
models which are adopted from the Cadence’s AHDL
library are also compared.
Figure 12. Vctrl Responses with Supply Voltage Variation
In order to check the detailed output responses, we
choose two arbitrary variation values (93.1mV and 38.2mV) produced by the random generator and observe
their corresponding results. We also compare several
critical parameters of PLL designs with the results of
HSPICE simulation in Figures 13 and 14. We can see that
the results of our behavioral models are still accurate under
supply voltage variation. Since the traditional behavioral
models cannot produce any output jitters as shown in
Figure 11, their results are not compared in the following
noisy conditions any more.
Figure 11. Comparison Results (Ideal supply voltage)
52
5. Conclusions
An efficient modeling approach with the effects of
supply voltage variation is presented in this paper to build
accurate PLL behavioral models in a short time by using
our bottom-up extraction flow. Because all the parameters
in our behavioral models are obtained from voltage-domain
measurement, their changes under supply voltage
fluctuation are almost proportional to the value changes of
supply voltage in our observation. Therefore, only three
post-layout simulations are enough to generate accurate
behavioral models under different supply voltages. This
approach can handle any distribution of supply voltage
variation and avoid time-consuming MC simulation.
Moreover, because no exact current values and device sizes
are required in our model, this approach will be more
suitable to verify the system performance with the effects
of non-ideal supply voltage while using existing IP designs.
The experimental results have shown that our behavioral
model can have very similar results to the HSPICE
simulation but require much less simulation time.
Figure 13. Results Comparison when dvdd = 0.0931V
Figure 14. Results Comparison when dvdd = -0.0382V
Finally, more random variation values are considered
and the corresponding output responses on period jitter and
lock time are shown in Figures 15 and 16. These two
parameters are highly sensitive to the supply voltage and
they are also the very important concerns of PLL designers.
When the supply voltage variation is increasing, our
behavioral models can still obtain similar lock time as in
the HSPICE simulations. About period jitters, the
simulation results of our behavioral models are also close
to the HSPICE results with the supply voltage fluctuation.
These results can show that we can build accurate
behavioral models for PLL designs under any given value
change of supply voltage.
6. References
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[2]
[3]
[4]
[5]
Figure 15. Agreement on Period Jitter
[6]
[7]
[8]
Figure 16. Agreement on Lock Time
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