Slew-Rate Effects in First Order Sigma

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IEEE MELECON 2004, May 12-15, 2004, Dubrovnik, Croatia
Slew-Rate Effects in First Order Sigma-Delta
ADC’s
Selçuk Talay, Günhan Dündar
Department of Electrical and Electronics Engineering, Bo aziçi University, stanbul, TURKEY
[email protected], [email protected]
Abstract—This work presents the effect of slew-rate on
first order, switched capacitor, Sigma-Delta (SD) analog-todigital converters (ADC). The slew-rate can be a significant
problem for SD ADC’s. Thus it should be related to the
input signal for accurate estimation of the error produced
by the slewing amplifier. In this work, the effects of the
signal histogram and its frequency spectrum have been
analyzed. Different cases have been presented in order to
illustrate the effect. Also, a model for estimating the error
caused by the slew-rate has been proposed.
helpful for further research into higher order systems.
Integrator
Input
Output
1-bit
ADC
1-bit
DAC
Figure 1. First order SD ADC.
The second section shows the effect of slew-rate in the
ADC. The following sections propose a model, which can
be used to estimate the slew-rate and develop more
efficient systems.
I. INTRODUCTION
Sigma-Delta ADC’s are gaining more and more
interest due to their various properties, which allow them
to be integrated into systems within various different
applications. They are suitable for many applications
ranging from medium speed telecommunication
applications to low speed, high resolution instrumentation
systems. One other advantage of this type of A/D
conversion is the small number of different blocks
required in their design. However, the number of possible
configurations that can be built with these blocks is quite
high [1] and designers commonly use only a few SD
architectures. There is a lot of research focused on
developing a SD design automation system that aims to
assist the designer throughout their design in order to
overcome this difficulty [1],[2],[3],[4]. However, systems
developed by some of these researchers are basically
behavioral simulators [2],[5] and they are far from being
a design automation system.
Even for behavioral SD simulation, the models
developed should be accurate enough for an efficient
design. Previously developed models are not interested in
slew-rate of the SD architecture and they rather prefer to
use slew-rate as a specification to be checked after a
design has been initially completed or throughout the
simulation. Although, this approach is still sufficient for a
successfully working SD ADC, it is not suitable for an
efficient design. The effect of slew-rate may lead to over
estimated components, which will unfortunately yield
inefficient designs such as selection of an OPAMP
consuming more power and area.
This work presents the slew-rate effect in first order
SD ADC’s. A first order SD block model is given in
Figure 1. The information presented here is believed to be
II. SLEW-RATE
Previous work on this topic [2], [3] mainly focused on
calculating the distortion caused by the slew-rate.
However, calculating the distortion is not sufficient. The
effects should be estimated for an ADC with a given
input signal. Thus more efficient designs can be possible.
Slew-rate is effective when the amplifier in the
integrator cannot supply sufficient current to the output.
As a result, for a switched capacitor integrator, a wrong
value of charge can be transferred to the feedback
capacitance. The small changes in the input signal do not
mean small changes at the input of the integrator; in other
words, the voltage, the symbol V in (1) is the voltage at
the integrator output.
SR=
d
V (t ) (V/ms)
dt
(1)
Cf
Vin
Φ1
Φ2
Cs
X +
Φ2
Φ1
-
Figure 2. First order, switched capacitor integrator.
In order to show this effect some brief information
about the switched capacitor integrator has been given in
the following paragraph.
Figure 2 shows the circuit diagram of a switched
capacitor integrator of a SD ADC [3]. The symbols Φi,
This work was supported by TÜB TAK by project
number 101E039.
0-7803-8271-4/04/$20.00 ©2004 IEEE
+
95
show two non-overlapping clock signals. The charge is
collected at the feedback capacitor Cf. The amplifier
should be “strong” enough to transfer the charge to this
capacitor. The input of the amplifier, node marked with
X, is an important node. The values at this node
determine whether the ADC enters a slew-rate condition
or not. In order to analyse the behaviour of the system a
MATLAB-Simulink model has been developed. Also, the
model of the non ideal integrator has been integrated into
Simulink which represents the nonidealities such as
capacitor mismatch, finite amplifier gain, gain error and
so on. The subblock that represents the slew rate of this
integrator is given in Figure 3.
However the information given in Table 1 is not
complete. The following sections will add more
information. Another important observation from the
analysis is the dependence on the frequency of the signal.
Although the slew-rate definition seems to be directly
related to the frequency, this is not the case in the
analysis. The frequency of the signal does not have
significant effect in the total error produced by the slew
rate. Different input signals have been given as an input
to the model. Results showing the effect of the frequency
are given in Figure 4 and Table 2.
TABLE II
MSE FOR TWO DIFFERENT FREQUENCIES
Period (number of samples)
Mean square error
Fig 4a
160 Samples
6.335e-006
Fig 4b
80 Samples
6.156e-006
1
Amplitude
0.5
0
-0.5
-1
150
200
250
1
300
Samples
350
400
450
500
Amplitude
0.5
0
-0.5
Figure 3. Slew-rate block of Simulink Model.
-1
The block in Figure 3 uses a linear model for slew-rate.
The block can be modified and the linear model can be
replaced by its more complex model such as tanh(x).
The developed MATLAB model was used to analyze
the behavior of the SD ADC with different input signals
that vary by their histogram and frequency spectrum. The
results showed that if the value of the input signal is
nearly half of the maximum input voltage, possibility of
errors due to slewing is maximum. In other words, if the
number of slewing conditions, which is the number of
samples when slewing errors occurs, has been observed
for various different DC voltages, it can be seen that the
values close to the mid level of the input range result in
higher number of slewing conditions. Table 1
summarizes some of these results for an input range from
–1V to 1V.
5.0%
0.1V
89.9%
-0.2V
79.8%
-0.9
5.0%
120
140
160
180
200
220
240
Figure 4 shows the inputs and the outputs, which are
converted back to analog by ideal DAC. This figure also
represents the effect of the frequency. Even though the
frequency has been doubled, the MSE has only changed
by less than 3%. These results are achieved when the
slewing condition is symmetric for rise and fall of the
signal. The results show that the frequency of the signal
does not effect the slew-rate distortion. Figure 5 shows
the effect of slew-rate with different slewing conditions.
The increase in the number of samples that the integrator
could not succeed in settling to its desired value, which is
also expressed as number of slewing condition, with the
frequency can be ignored. The total difference, as shown
in Figure 5, is less than 4% for 200% increase in input
signal frequency.
Figure 5 shows the ratio between the number of error
conditions to the total number of samples (RE/T) for
different slewing conditions (RC) when the frequency is
increased 8 times. The horizontal axis shows the ratio
between the output voltage that OPAMP can supply
within a sample time and maximum voltage possible at
Percent of slewing conditions
0.9V
100
Samples
Figure 4. Ramp input for two different frequencies showing the
ideal input, ADC output which is slewing and the error between
them.
TABLE I
INPUT LEVELS AND SLEWING CONDITIONS
Input Level
80
96
0.4
Figure 1 shows the block diagram of the first order SD
ADC. If the input is x[n], output y[n], input of the
quantizer w[n] the following equations can be written:
RE/T
0.3
y[n]=sgn( w[n] )
w[n]= w[n-1] + x[n-1] – y[n]
0.2
0.1
which leads to:
0
1
0.9
0.8
0.7
w[n] - w[n-1] = x[n-1] – sgn( w[n] )
0.6
RC
(RE/T) for 5 different slewing conditions for increased frequency.
node X in Figure 2. Each of the five curves represent
different frequencies where the lower values correspond
to slower OPAMP’s and the value 1 on the x-axis stands
for situations in which the OPAMP will never slew.
The number of slewing conditions is very important.
The distortion at the output of an ADC can be represented
by MSE of the error, where error is the difference
between the ideal signal and the converted signal as
illustrated in Figure 4. On the other hand it is easier to
estimate the slewing condition from the model. The
square of the number of slewing conditions is actually
linearly proportional to MSE. This relation can be
observed from Figure 6. The horizontal axis shows the
square of the ratio that is the number of slewing condition
occurred over the total number of samples. Three
different curves represent different slew rate conditions.
-6
x 10
MSE
2.5
2
1.5
1
0.5
0
0
0.02
0.04
0.06
0.08
(3)
The left side of the expression in (3) actually shows the
difference of the current output of the integrator with the
previous output, which causes slewing condition. If the
difference is larger than the corners defined in linear
slew-rate characteristic, slewing condition occurs. This
equation also represents the reason why mid level inputs
may cause more slewing conditions. The output of the
DAC, which has been represented by the signum
function, will have nearly equal number of –1 and +1 for
medium level signals and is similar to a square wave
oscillator. Thus, smaller x[n] values will bring large
number of slewing conditions in a system which has
lower slew-rate limits. However, if input has values
closer to the limits the DAC output will be either –1 or +1
most of the time. Since the negative feedback tries to
compensate, the input and the DAC output will have
opposite signs. Thus, the difference between them will
bring smaller differences most of the time. However, it
will also present a large difference for small amount of
time. The simulations have shown that it was 5% of the
total samples as it was stated in Table I. Also, slew-rate
requirement becomes more stringent. This is the case
when both input and the DAC output have the same sign.
The information presented here shows that for a
system which has a signal histogram having peak values
at center and which has a lower slew-rate level will have
large number of slewing conditions. This type of input
signals requires careful amplifier design. The slew-rate
capacity should large enough. Also capacitor values
should be selected as small as possible to relax the slewrate requirements. On the other hand, if the signal
histogram has values spread all over the input range or
having its peak values near the limits, the system may
tolerate smaller slew-rate values. Although, this approach
will bring error to the system it can be tolerated for mid
level resolutions. Depending on the application, this
approach may relax the tight limits on the amplifier
specifications.
Another important result of this analysis is that, the
sinusoidal inputs may not be the correct input for testing
the slew-rate effect on SD ADC. Since the histogram of
the sinusoidal input is mainly collected at the limits of the
input spectrum, possibility of slew-rate conditions is very
small. More suitable input will be a signal with Gaussian
like histogram.
Figure 5. Ratio of error conditions to the total number of samples
3
(2)
0.1
(RS/T)2
Figure 6. MSE vs. square of the ratio of slewing conditions to the
total sample number.
III. SLEW-RATE MODEL
Previous sections show the simulation-based
information. This information can be obtained from
analytical expressions. For a system where the reference
levels are –1V and 1V, the DAC output can be expressed
by a signum function. Thus, by using the discrete time
definitions, an expression for the node X in Figure 2 can
be found. However, this expression should be in terms of
input such that it allows us to estimate the slewing
condition from the input. If the histogram of the input
signal is known, the total number of slewing occurrences
can be estimated which can lead to the estimation of total
error that will add up at the output.
IV. EXAMPLE
The number of slewing conditions should be defined in
terms of input signal in order to estimate the number of
97
taking the value that causes slewing conditions within the
input range affecting the slewing is 39%. This is very
close to the real value, 40%.
In order to test the model some other inputs were
applied. The above example is a synthetic example and
may not represent real life signal characteristics. Thus, a
speech file was given as input. The results showed that
the analytical calculations yielded an error less than 5%.
Also, some other input signals such as sum of several
sine waves were applied to the model and the error
between the actual slewing condition number and the
calculated slewing condition number was observed to be
between 1% to 3% for all different inputs.
These examples show that this approach is suitable for
quite accurate estimations of slewing conditions, which
may be used by designers. Since this approach is only
related to the input histogram and DC level of the signal,
the calculation time is quite small.
slewing conditions. This information then, can be
converted to the MSE value with the conversion
presented in Figure 6. Although the expression given in
(3) summarizes the relation, some numerical results will
clarify the situation.
For an input signal limited by –1V to +1V, the
maximum difference at the node X is basically the
maximum of the input signal ±1 from (3); that is if the
maximum value of the input signal is ±0.4V, the
maximum difference will be –1.4V and +1.4V. For an
example design, which allows up to 1.12V per each
transition, (3) gives the following values:
2.12V
0.12V
(4)
-2.12V
-1.12V= xlim ±1 =
-0.12V
These limits determine where the slewing errors start.
For input signal values exceeding 0.12V and –0.12V,
slewing errors start, which gives a coarse estimate of the
number of slewing conditions. For input samples with
values greater than ±0.12V such as 0.3V, there are two
possibilities according to the digital output. One is 1.3V
and the other is –0.7V. Only 1.3V will introduce a
slewing condition. For values such as 0.1V there will be
no slewing. If the output node has equal possibility of
taking –1 or +1 values, which is a coarse estimation, the
number of slewing conditions will be equal to the half of
the number of samples having values greater than
±0.12V. For our example, the total number of samples is
8192, and number of samples with smaller values than
±0.12V is 2604. If the possibility of the output to take the
value, which causes the slewing condition, is 50%, the
total number of slewing conditions should be 2794. The
MATLAB simulations showed that the real value for
slewing conditions is 2301. The error in this approach is
nearly 20%. Although this value may give an opinion on
the order of magnitude of the problematic situations, the
number should more accurate. The reason of this problem
is basically taking the possibility of the digital output as
50%. If the possibility is 50%, this means that the average
of the input is 0, since the digital output actually shows
the average of the input signal. In the given example, it
can be observed that the possibility of values causing
slewing conditions is less that 50%, actually 40%. Since
the limit of the effective range has been defined and the
average value of the input signal can be calculated in
order to get the ratio of ±1 values at the output, an
accurate estimation can be calculated. For the example
presented in this section the input signal is a sawtooth
signal with ±0.4V of maximum values. The possibility of
1.12V= xlim ±1 =
V. CONCLUSION
In this work we presented the effect of slew-rate on
first order switched capacitor, SD ADC. Also analytical
expressions were given in order to clarify the effect.
Results selected from various simulations have also been
presented.
The effect of slew-rate can be very problematic for
some applications and if the designer has the required
information about the input signals, more efficient
circuits can be designed. The information presented here
can also be helpful for researchers developing design
automation systems for ADC’s [6].
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F. Medeiro, B. Perez-Verdu and A. Rodriguez-Vazquez, TopDown Design of High-Performance Sigma-Delta Modulators,
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