Integration of Frequency Response Measurement Capabilities in Digital Controllers for DC–DC Converters

Integration of Frequency Response Measurement Capabilities in Digital Controllers for DC–DC Converters
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 23, NO. 5, SEPTEMBER 2008
Integration of Frequency Response Measurement
Capabilities in Digital Controllers
for DC–DC Converters
Mariko Shirazi, Jeffrey Morroni, Student Member, IEEE, Arseny Dolgov, Regan Zane, Senior Member, IEEE, and
Dragan Maksimovic, Senior Member, IEEE
Abstract—Recent work has shown the feasibility of integrating
nonparametric frequency-domain system identification functionality into digital controllers for switched-mode pulse-width
modulated (PWM) dc–dc power converters. The resulting discrete-time frequency response can be used for design, diagnostic,
or self-tuning purposes. The success of these applications depends
on the fidelity of the identified frequency responses and the degree to which the process is automated, as well as the costs, in
terms of gate count, time duration of identification, and effect
on output voltage, incurred to obtain these benefits. This paper
demonstrates the feasibility of incorporating fully automated
frequency response measurement capabilities in digital PWM
controllers at relatively low additional cost. In particular, it is
shown that relatively accurate and smooth frequency response
data can be obtained using a Verilog-coded implementation with
low tens of thousands of logic gates and about 10 kB of memory.
The identification process can be accomplished in several hundred
milliseconds and the output voltage can be kept within specified bounds during the entire process. Experimental results are
provided for four different PWM dc–dc converters, including a
synchronous buck with two different filter capacitors, a boost
operating in continuous conduction mode (CCM), and a boost
operating in discontinuous conduction mode (DCM).
Index Terms—Binary sequences, correlation, dc–dc power
conversion, digital control, frequency response, Hadamard transforms, identification, pulse-width modulated (PWM) power
converters, quantization, switched mode power supplies.
I. INTRODUCTION
IGITALLY controlled switching power converters have
the potential to offer increased functionality and performance relative to traditional control methods due to the ease
with which complex, intelligent, and/or adaptive algorithms
can be implemented in the controller at low additional cost
[1]. Recent work in digital control for pulse-width modulated
(PWM) dc–dc converters [2]–[6] has shown the feasibility of
integrating nonparametric frequency-domain system identification functionality. The resulting discrete-time frequency
D
Manuscript received December 20, 2007; revised March 06, 2008. Current
version published November 21, 2008. Recommended for publication by Associate Editor H. Chung.
The authors are with Colorado Power Electronics Center, Department of Electrical and Computer Engineering, University of Colorado, Boulder, CO 80309
USA (e-mail: [email protected]; [email protected];
[email protected]; [email protected]; [email protected]).
Digital Object Identifier 10.1109/TPEL.2008.2002066
response can be used for design, diagnostic, or self-tuning
purposes. The success of these applications depends on the
fidelity of the identified frequency responses and the degree to
which the process is automated, as well as the costs, in terms of
gate count, time duration of identification, and effect on output
voltage, incurred to obtain these benefits.
System identification generally falls into two main categories: parametric and nonparametric methods [7], [8].
Parametric methods return the parameters of the system model
such as the coefficients of a system difference equation, transfer
function, or state-space model. Nonparametric methods return
impulse response and/or frequency response data directly. The
distinction is sometimes blurred for those applications that
apply parametric methods to the results of a nonparametric
analysis, e.g., [9], [10]. Parametric methods require, in addition
to selection of an appropriate input stimulus, a priori selection
of a parameterized model structure including system order and
number of zeros, construction of a suitable prediction error
equation and loss function, and methods to minimize the loss
function. These methods have been applied to switched mode
power converters experimentally [10]–[15] and through simulation [9], [16], [17], and are useful, e.g., for more complex
controller design [10]. Nonparametric methods do not assume
a system model and require only selection of an appropriate
stimulus.
Nonparametric identification methods include transient response, cross correlation, frequency response, Fourier analysis,
and spectral analysis [7], [18]. Transient-response methods,
which inject either a pulse or step input, require large input and
output perturbations to obtain good signal-to-noise ratios [7].
The frequency response methods, which excite the system with
pure tones, require long identification times to obtain results
over a wide range of frequencies with good frequency resolution [7], [19]. Alternatively, in the Fourier analysis method, a
multifrequency input is applied and the frequency response of
the system is computed as the Fourier transform of the output
divided by the Fourier transform of the input [7]. This method
requires complex division at each frequency. In a related
approach using spectral analysis, the frequency response is
computed as the cross-spectral density estimate between input
and output divided by the power spectral density estimate of
the input [7], [8], [18]. The cross correlation method exploits
the fact that for a white input (i.e., one with zero mean whose
autocorrelation function is equal to a pulse at the origin and
whose power density spectrum is flat), the impulse response of
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SHIRAZI et al.: FREQUENCY RESPONSE MEASUREMENT CAPABILITIES IN DIGITAL CONTROLLERS FOR DC–DC CONVERTERS
Fig. 1. Digitally controlled PWM converter with integrated frequency response
measurement capabilities.
the system is proportional to the cross correlation between input
and output, while the correlation itself rejects any disturbances
to the system as long as they are uncorrelated with the input
[2]–[8], [11], [18]–[25]. Correlation analysis, followed by a
fast Fourier transform (FFT) to obtain the frequency response,
is identical to the spectral analysis method using a white input.
The most commonly used input stimulus for correlation analysis is a pseudorandom binary sequence (PRBS) generated as a
maximum-length sequence (MLS). Correlation analysis using
MLS has been used for decades by audio engineers to measure
acoustical systems [19], [23]–[33], with more recent applications to power electronics experimentally [2]–[6], [11], [21] and
in simulation [22]. None of the audio works, and only [5] and [6]
of the power electronics works, have implemented the complete
identification using hardware description language (HDL).
In practical applications of correlation-based identification
to digitally controlled power converters, the analog-to-digital
converter (ADC) introduces quantization error that corrupts
the identification results. Over the years, the audio engineering
community has developed techniques to improve noise and
distortion immunity of MLS-based transfer function measurements. These methods, which will be discussed in more
detail later, include preemphasis/postemphasis [19], [23], [25],
[31], [42], impulse-response truncation [23], [27], [29], and
fractional-decade smoothing [26], [33].
This paper presents a practical, hardware-efficient implementation of correlation-based system identification for PWM
dc–dc power converters, adapting methods from the audio
engineering field to reduce the effects of ADC quantization to
obtain an accurate and smooth system frequency response. The
approach is completely automated online and can be applied
to a wide range of PWM dc–dc converter architectures with
no changes to the identification algorithm. An overall block
diagram of the proposed system is shown in Fig. 1. Identification can be enabled at discrete instances during operation,
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e.g., upon start up or following a change in load. During
identification, the system is run in open loop, with the comfrozen at its steady-state value. The
pensator output
system identification block generates the PRBS,
, and
shapes it with a preemphasis filter to reduce the effects of ADC
, is injected into
quantization. The resulting sequence,
.
the digital pulsewidth modulator (DPWM) on top of
The shaping is removed by deemphasis of the measured output.
Cross correlation to obtain the impulse response is efficiently
computed using the fast Walsh–Hadamard transform (FWHT),
followed by truncation and FFT. Finally, the FFT data are
smoothed with a fractional-decade spectral window.
The paper begins with a review of previous work in system
identification by cross correlation in Section II, while Section
III presents a hardware-efficient implementation using fully
synthesizable Verilog code. Section IV shows the effects of
ADC quantization and presents the automated implementation
of preemphasis/deemphasis and the spectral smoothing techniques of impulse response truncation and fractional-decade
spectral smoothing. Section V presents the method implemented to automate selection of the PRBS magnitude. Finally,
experimental results for several PWM dc–dc converter topologies incorporating these methods are presented in Section VI.
II. REVIEW OF THE CROSS CORRELATION METHOD
The method of identification using cross correlation is well
known, e.g., [2], [7], [20]. The salient result and application
to a switched-mode power converter is presented here, with
reference to Fig. 1. A switched-mode power converter can be
regarded as a discrete-time linear time-invariant system if the
system is in steady-state and the input perturbations are small.
is chosen as the input stimulus
If white noise with variance
, then the cross correlation of the input signal with the
output error signal
is given by
(1)
is the discrete-time system impulse response. The
where
converter control-to-output transfer function can then be found
.
by applying the discrete Fourier transform (DFT) to
From a practical standpoint, it is convenient to approximate
the white noise input with a PRBS based on an MLS. A PRBS
is a deterministic periodic signal that can be easily generated
in hardware using a shift register and feedback taps [34]–[38].
For a p-bit shift register, the location of the taps can be selected so that the period of the resulting sequence of 0's and
. The corresponding
1's has the maximal length of
in Fig. 1, can be obtained by replacing the 0's
PRBS,
with +ū and the 1's with ū in the MLS, where ū is the desired
duty-cycle perturbation magnitude. The PRBS length must be
designed such that the impulse response of the converter decays within one injection period of the PRBS to ensure that the
complete response is obtained. If this requirement is met, then
output data, sampled once-per-switching cycle, can be collected
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 23, NO. 5, SEPTEMBER 2008
=0
= 15
= 100
= 700 Fig. 2. Simulated identification of CCM boost with q
. (a) Power stage parameters: V
V, L
μH, R
m , and switching
kHz. (b) Impulse response. (c) Frequency response. In plots (b) and (c), the identification results (gray dots) are plotted against the discretefrequency, F
time model of the converter (black line).
= 195
over one period of the PRBS only and the cross correlation can
be made circular so that (1) can be arranged as
1)-point input and output sequences, zero-padded to N-point sequences, sampled at the switching frequency .
III. HARDWARE EFFICIENT IDENTIFICATION
(2)
The circular cross correlation of (2) assumes that the input
and output are periodic. Thus, it must be ensured that the system
has reached periodic steady state with respect to the input PRBS
before the output measurements are taken. If the PRBS length
has been appropriately chosen, it is sufficient to inject the PRBS
twice and collect the output data over the second injection period. Even then, (2) is strictly valid only if any disturbances to
the system are uncorrelated with the input and are either periodic having a frequency which is an integer multiple of that
of the PRBS or are transient and decay within the PRBS period. Under these conditions, the circular cross correlation will
give the exact discrete-time impulse response (reversed in time
modulo-N). This is verified by simulation of the continuous conduction mode (CCM) boost converter shown in Fig. 2(a). The
simulation was performed in Simulink with no noise sources or
ADC nonlinearities modeled in the system. The resulting identification produces the exact converter impulse response and after
zero-padding (to make the length equal to a power of 2) and performing the DFT, its frequency response exactly matches the
, as shown in
discrete-time model prediction [39] up to
Fig. 2(b) and (c). The frequency samples of the resulting DFT
Hz apart, corresponding to the (N –
are spaced
The cross correlation between the two sequences of (2) is a
operations.
matrix multiplication that takes
However, the MLS has certain symmetry properties that allow
for efficient computation of the cross correlation function [3],
[24], [40]–[44]. The steps to exploit this symmetry begin with
the augmentation of the matrix multiplication upon which (2) is
based to obtain
..
.
..
.
..
.
..
.
..
.
..
.
(3)
where ũ[n] is the MLS with 0's replaced with +1 and 1's replaced
with 1 and ū is the magnitude of the PRBS. The transformation
represented by (3) is called the M-transform. The matrix of the
and can be shown to be perM-transform has order
mutationally equivalent to a Hadamard matrix of the same order
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SHIRAZI et al.: FREQUENCY RESPONSE MEASUREMENT CAPABILITIES IN DIGITAL CONTROLLERS FOR DC–DC CONVERTERS
[40]–[43] according to
, where M is the matrix of
the M-transform, H is the Hadamard matrix of the same order,
and
are the permutation matrices to reorder the rows
and
and columns of H, respectively. Thus, (3) becomes
(4)
with
and ê*[n] representing the vectors
and e*[n] augmented according to (3).
The parenthetical structure of (4) reveals how the effects
and
can be realized by a simple reordering of the
of
rows of the vectors they operate on. Thus, the M-transform has
,
been effectively replaced by a transform of the form
called the Walsh–Hadamard transform (WHT). At first glance,
it seems that nothing has been gained, as the WHT of (4)
operations. However, the WHT belongs
now requires
to the class of generalized Fourier transforms, and thus, there
exists a fast transform, the FWHT, which is based on the same
butterfly structure as the well-known FFT and which reduces
operations. Finally, then, the
the order of the transform to
operations of the matrix multiplication of (2) have
been replaced by the following steps.
1) Zero-pad and then reorder the rows of e*[n] according to
.
operations).
2) Perform the FWHT on the reordered data (
3) Reorder the transformed data according to
.
4) Drop the zeroth term of the result and reverse in time
.
modulo-N to obtain
For p = 10, this method replaces 1 M additions with 10 k additions and two reorderings. Furthermore, as shown in [28],
[40], and [44], the permutation vectors that define the permuand
can be easily generated on-the-fly
tation matrices
using shift registers uniquely defined by the MLS upon which
the input is based, thus eliminating the need to store them in
memory.
to obtain
The final step is to take the FFT of
. Since the FWHT is identical to the FFT with
multiplication by complex exponentials replaced by additions
and subtractions, the two algorithms share the same code. In
addition, although the FFT requires a complex exponential
registers long due to
lookup table (LUT), it need only be
symmetry.
IV. PREEMPHASIS/DEEMPHASIS AND SPECTRAL
SMOOTHING TECHNIQUES
Section II presented simulated identification results for an
ideal system with no external noise sources and an infinite resolution ADC. In practical applications, the ADC of the digitally controlled power converter introduces quantization error
that compromises the fidelity of the frequency response identification. In particular, since the converter is a low-pass filter,
the signal-to-noise ratio of the measured output can be considerably attenuated at high frequencies. Thus, for any converter,
the dynamic range of the identification process, and therefore,
the range of frequencies over which the identification is valid,
increases with the input perturbation and ADC resolution. However, the input perturbation magnitude is limited by the specified
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output voltage tolerances of the converter. In particular, a stringent regulation band (e.g., ±3%) of the nominal output voltage
is common for low-voltage buck converters. Furthermore, as the
goal of this paper is to present a practical implementation, a
, is assumed.
modest ADC quantization interval,
The system shown in Fig. 2(a) was resimulated with an effective (prior to sensing network) ADC quantization interval of
262 mV (0.9% of the output voltage), and an input duty-cycle
perturbation magnitude of ±1.0% (which results in an output
voltage perturbation of ±3%). As expected, the simulated identification results, shown in Fig. 3(a) and (c), are corrupted by
quantization error. Fig. 3(b) and (d) shows experimental results
for the same system and perturbation magnitude, with output
voltage sampled between 0.7 and 0.8 μs prior to the gate-on transition to avoid sampling in the presence of switching noise. The
experimental results closely match the simulated ones. Since no
other nonlinearities or noise sources were included in the simulation, it is clear that the corruption in the experimental results
is dominated by quantization error.
A. Preemphasis/Deemphasis
Fig. 3 shows the severe degradation of identification results
due to reduction in signal-to-noise ratio at high frequencies. A
similar problem is encountered in FM broadcasting and audio
recording systems and is solved using preemphasis/deemphasis
methods [32], [45]. The basic concept is to spectrally shape,
or preemphasize, a signal prior to injection into a system in a
way that makes it less susceptible to noise introduced by the
system, and then, to apply the inverse filter to, or deemphasize,
the output of the system to remove the shaping. This technique
has also been used in the audio engineering field to improve
the signal-to-noise ratio in identification experiments [19], [23],
[25], [31], and is adapted here, with extensions to allow automated design of the preemphasis filter.
The preemphasis/deemphasis process employs two filters.
The preemphasis filter takes as input the original PRBS,
, and outputs
, which is a multilevel sequence
with high-frequency boost. If the preemphasis filter has been
properly designed, the resulting high-frequency components of
, will have been boosted sufficiently
the measured output,
above the noise floor introduced by the ADC converter to
significantly reduce the effects of quantization error. The deemphasis filter is the exact inverse of the preemphasis filter and
to eliminate the shaping introduced by
must be applied to
the preemphasis filter so that cross correlation of the resulting
, with the unfiltered PRBS input,
, will
signal,
.
return
The preemphasis filter chosen is a simple first-order finite-impulse response (FIR) filter with the following template transfer
function:
(5)
Here, K is the filter gain and
is the filter corner frequency.
For a given PRBS magnitude, the filter gain K and corner freare selected according to the following criteria.
quency
1) The filter must boost the output above the noise floor introduced by the ADC.
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 23, NO. 5, SEPTEMBER 2008
= 262
Fig. 3. Simulated and experimental identification of CCM boost of Fig. 2(a) with effective q
mV. (a) and (c) Simulated impulse response and frequency
response. (b) and (d) Experimental impulse response and frequency response. In all plots, the identification results (gray dots) are plotted against the discrete-time
model of the converter (black line).
2) The filter must not saturate the duty cycle.
3) The preemphasized input must not cause the output voltage
to exceed tolerances.
low enough
It is possible to select K large enough and/or
so that criterion (1) would be satisfied for almost any converter
desired. However, this would invariably cause criteria (2) and/or
(3) to be violated in many cases (in fact, the same could have
been achieved simply by increasing the PRBS magnitude). If
the original PRBS magnitude is chosen to maximize the output
voltage perturbation subject to the specified regulation band, it
is desired to design a filter that provides the minimum amount
) necof emphasis (in terms of the size of K and location of
essary to satisfy criterion (1). One way to achieve this is to place
at the frequency where it is expected that quantization error
will begin to dominate the signal. This frequency depends not
, but also on the roll-off
only on the PRBS magnitude and
characteristics of the converter being identified. The filter design method implemented here is to first perform identification
without preemphasis and use the resulting frequency response to
design the preemphasis filter. If the frequency response results
shown in Fig. 3 are smoothed using methods to be described
can be selected as that frequency
in the next section, then
Fig. 4. Experimentally identified impulse response of CCM boost of Fig. 2(a)
with effective q
of 262 mV and preemphasis/deemphasis.
where the magnitude of the frequency response drops below
where
is the PRBS magnitude.
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SHIRAZI et al.: FREQUENCY RESPONSE MEASUREMENT CAPABILITIES IN DIGITAL CONTROLLERS FOR DC–DC CONVERTERS
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Fig. 5. Complete identification process with preemphasis/deemphasis and spectral smoothing.
With the filter corner frequency selected as before, it is desired to select K so that the preemphasis filter has a dc gain of
one. However, it must also be ensured that this value does not
saturate the duty cycle. This can be done using a result from robust control (see, e.g., [46])
(6)
Fig. 6. CCM buck converter hardware prototype.
which states that for a linear shift-invariant system with impulse
norm of the output y[n] is less than or equal
response f[n], the
norm of
to the norm of the impulse response times the
,
the input u[n]. For the case here with the PRBS input
, and output
,
first-order high-pass FIR filter
(6) becomes
(7)
The derivation of (7) from (6) consists of three steps. First,
is binary, its
norm is simply the PRBS magnisince
tude,
. Second, the inequality in (6) can be replaced by
equality according to [47] in which it is shown that the limit is
achieved when u[n] is a binary input whose sign matches that of
f[n] flipped in time. For the first-order FIR filter of (5), all that
go from negative to posis required for equality is that
itive anywhere in the sequence. This is the case for any PRBS
whose value changes more than once. Finally, it can be easily
shown that for an FIR filter with all zeros on the positive real axis
,
(high-pass filter),
where
, is the
norm of the filter. The result
is that (7) can be used to exactly compute the maximum value
of K to not saturate the duty cycle, i.e., to meet criterion (2).
It should be noted that there are some systems for which the
benefits achieved using preemphasis/postemphasis may not be
practically realizable. This may happen, for example, with a
buck converter with very low resonance (e.g., only several times
f) and a high zero from the capacitor equivalent series resistance (ESR). The slow response of this converter means that it
will require a large duty cycle perturbation, even during unfiltered PRBS injection, to achieve significant output voltage perturbation. The large dynamic range (the span in decibel from the
largest magnitude response, typically at dc or near resonance,
) of the
to the smallest magnitude response, typically near
converter means that the preemphasis filter corner frequency
will be quite low. These two factors combined would ideally
result in large duty cycle perturbations during preemphasized
injection. However, if, in addition, the converter operates with
a very low or very high steady-state duty cycle, the duty cycle
perturbations will be limited by saturation, and it is possible that
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Fig. 7. Comparison of the incremental effect of processing techniques on the experimentally identified frequency response of the CCM boost of Fig. 2(a). (a) Pre-/
deemphasis only. (b) Fractional-bandwidth smoothing only. (c) Pre-/deemphasis and fractional bandwidth smoothing. (d) Pre-/deemphasis, fractional-bandwidth
smoothing, and impulse-response truncation. In all plots, the identification results (gray dots) are plotted against the discrete-time model of the converter (black
line).
preemphasis may not improve the fidelity of the identified frequency response at all. The large filter action may, in fact, excite
nonlinearities of the DPWM, resulting in distortion of the frequency response.
B. Spectral Smoothing Techniques
Fig. 4 shows the experimental impulse response obtained
from identifying the same converter of Fig. 2(a) with the
addition of preemphasis/deemphasis filtering based on the
automated design procedure detailed earlier. It can be seen that
even with preemphasis/deemphasis, the impulse response still
contains low-level noise distributed along its length. The audio
engineering field has shown that this noise often consists of
both random noise as well as fixed-pattern noise, or distortion,
introduced by nonlinearities. Furthermore, for a sufficiently
long injection period, the linear portion of the impulse response
will be entirely contained in the first part of the measured impulse response, while the tail will consist entirely of the noise
components [23], [27], [29]. Thus, truncation of the impulse response, prior to taking the FFT, increases both the random noise
and distortion immunity of the resulting frequency response,
and is preferred over simple averaging techniques that would
improve random noise immunity, but not distortion immunity
[23], [27]. A further benefit of impulse response truncation is
that if the PRBS length is properly chosen, it can be used to
eliminate distortion artifacts in the impulse response arising
from DPWM nonlinearities, similar to observations noted in
[29].
While impulse response truncation provides some spectral
smoothing, even greater benefits are achieved by directly
smoothing the frequency response by convolving the FFT
data with a spectral window. The window should be narrow
enough not to smear true resonant peaks (thus introducing bias
into the frequency response), but wide enough to sufficiently
smooth roughness caused by spurious peaks [7]. In fact, the
optimum window is frequency dependent—narrow over frequency ranges containing resonant peaks (i.e., frequencies near
and below the resonant peak of a low-pass filter) and wide over
frequency ranges containing many spurious peaks (i.e., high
frequencies) [7]. To achieve this, fractional-decade spectral
smoothing, which uses a constant relative bandwidth versus
constant bandwidth, spectral window can be applied, as shown
in audio applications [26], [33]. A common choice of window
size in audio applications is one-third octave smoothing, i.e.,
a half-window width of one-third octave, as this is the range
spanned by the human ear's critical bands [26]. However,
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SHIRAZI et al.: FREQUENCY RESPONSE MEASUREMENT CAPABILITIES IN DIGITAL CONTROLLERS FOR DC–DC CONVERTERS
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= 600 μF.
Fig. 8. Experimentally identified frequency responses of the converters of Figs. 2(a) and 6. (a) CCM boost. (b) DCM boost. (c) Buck with C
: mF. In all plots, the identification results (gray dots) are plotted against the discrete-time model of the converter (black line).
(d) Buck with C
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for pure identification purposes, i.e., in order to obtain the
smoothest but still unbiased frequency response estimate, it is
apparent that the bandwidth fraction should itself be dependent
on the characteristics of the system being identified. In particular, the bandwidth fraction should be inversely proportional to
the frequency at which the magnitude transfer function begins
to roll off, as this is where the signal-to-noise ratio also begins
to decrease. The method implemented here is to set the band, where f is the frequency
width fraction equal to
, and
is the
resolution of the identification, equal to
frequency where the phase of the frequency response drops
below 90 , easily identified as the first frequency where the
real part of the frequency response becomes negative. Since
will occur at a frequency prior to
for any stable system
substantial attenuation, it can be identified from unsmoothed
results.
In order to implement fractional-octave smoothing, [26] interpolates the frequency response data to produce data points
equally spaced along a logarithmic frequency axis, smooth these
data by convolution with the desired fractional-octave window,
and then, performs an inverse interpolation to produce smoothed
data equally spaced along a linear frequency axis. The method
adopted here is to simply perform a moving average on the linearly spaced frequency response data using a dynamically sized
window that increases with frequency.
Fig. 5 shows the complete identification process with preemphasis/deemphasis as well as spectral smoothing by impulse response truncation and fractional-bandwidth smoothing.
V. AUTOMATED SELECTION OF PRBS MAGNITUDE
The preemphasis filter design presented in Section III assumes that the PRBS magnitude has been chosen to maximize
the output voltage perturbation subject to the specified regulation band. For a given ADC quantization level, the dynamic
range of the identification will increase as the PRBS magnitude
is increased, until the point at which the converter is driven out
of its linear operating range. Under the reasonable assumption
that the specified regulation band of the converter is such that the
converter operation remains largely linear over this range, it is
desired to select the PRBS magnitude to achieve the maximum
allowable output voltage deviation. The magnitude required to
achieve this depends on the converter itself, and therefore, it is
desirable to automate its selection online. The procedure implemented here is to, starting with the smallest possible PRBS
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Fig. 9. Output voltage during identification of converters of Figs. 2(a) and 6. (a) CCM boost. (b) DCM boost. (c) Buck with
C
: mF. In all plots, the injection enable signal is plotted below the output voltage waveform.
=26
magnitude of 1 LSB of DPWM command,
, linearly increase the magnitude until an output voltage deviation of at least
is recorded, and then, to continue to increase the magni2
tude until the first point at which the output voltage deviation increases further. This provides a range of PRBS magnitudes over
which the quantized output voltage deviation remains constant.
The midpoint of this range is then selected and scaled by the
ratio of the specified regulation band to recorded output voltage
deviation to achieve a PRBS magnitude that will result in output
voltage deviation close to the regulation band.
VI. EXPERIMENTAL RESULTS
The CCM boost converter described in Section II as well as a
discontinuous conduction mode (DCM) boost converter and two
CCM buck converters were experimentally tested with the automated preemphasis/deemphasis and spectral smoothing techniques described in Section V. All converters were operated
C
= 600 μF. (d) Buck with
at a switching frequency of 195 kHz. The parameters of the
DCM boost converter are the same as in Fig. 2(a), except for
μH,
m , and
V. The experimental buck converter power stage is illustrated in Fig. 6. This
system was identified with two different output capacitances,
μF and
mF. Output voltage was
sampled once-per-switching cycle just prior to the gate-on transition with an effective ADC resolution of 262 mV for the boost
cases and 7.8 mV for the buck cases. The digital controller and
system identification functions were implemented on a Xilinx
Virtex-IV field-programmable gate array (FPGA). A 10-bit shift
register (p = 10) was used to generate the PRBS sequence, resulting in a period of 1023 points and a frequency resolution of
190 Hz. Four 1024 × 18 bit RAM blocks were used to perform
and store the results of the cross correlation and FFT. All calculations were performed using fixed-point arithmetic. For the
FFT computations, 18 × 18 bit multipliers were used, 16 × 16
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SHIRAZI et al.: FREQUENCY RESPONSE MEASUREMENT CAPABILITIES IN DIGITAL CONTROLLERS FOR DC–DC CONVERTERS
2533
TABLE I
SUMMARY OF IDENTIFICATION RESULTS
TABLE II
RESOURCE REQUIREMENTS
bit multipliers were used for the preemphasis filter, and 22 × 16
bit multipliers were used for the postemphasis filter. The 18-bit
wide RAM blocks and FFT multipliers resulted in FFT magnitudes that were very similar to the floating point results obtained
in Matlab with the same output voltage data. For example, for
the 600 μF buck case, the worst case error between fixed-point
and floating-point results, which occurs at the lowest identified
frequency of 195 Hz, was 22 dB. The error decreased rapidly
from this point, with an error at resonance of 42 dB. If such
precision is not required, 16-bit wide RAM blocks and FFT multipliers may be sufficient.
Fig. 7 shows, for the CCM boost, the incremental improvements seen by using the preemphasis/postemphasis, fractionalbandwidth smoothing, and impulse response truncation techniques of Section III. There are noticeable improvements when
using each method individually, but the smoothest and most accurate frequency response is obtained by using the combination
of techniques.
Fig. 8 shows the identification results using all three processing techniques (preemphasis/deemphasis, fractional bandwidth, and impulse response truncation) for all the converter
topologies outlined before. In Fig. 9, the time-domain waveforms of the output voltage for each converter are also included,
showing the automated sweep to determine PRBS magnitude,
injection period for the identification without emphasis, computation time, and injection period for the identification with
emphasis.
A summary of the system identification results, including
PRBS magnitude, preemphasis filter parameters, output voltage
perturbation magnitudes, and total identification duration, as
well as the specified regulation bands for each converter is given
in Table I. It can be seen that although the preemphasis filter
does in some cases increase the output voltage perturbation, the
and could in practice be accounted
increase is only one
for by reducing the allowable output voltage tolerance used to
select the PRBS magnitude. Finally, Table II lists the resource
requirements, in terms of number of logic gates and required
memory, to implement the identification core as well as the automated magnitude selection, preemphasis/deemphasis, and spec-
tral smoothing functions. It should be noted in this table that
the sum of the logic gates for each function is greater than the
total logic gates required. This is due to resource sharing. In
particular, the automated magnitude selection and the spectral
smoothing functions share a 4100-gate divider.
VII. CONCLUSION
This paper has demonstrated the feasibility of incorporating
fully automated frequency response measurement capabilities
in digital controllers for PWM dc–dc converters at low additional cost. In particular, it has been shown that relatively
accurate and smooth frequency response data can be obtained
without requiring a high-resolution ADC or large output voltage
perturbations through the adaptation of preemphasis/postemphasis, fractional-bandwidth smoothing, and impulse response
truncation techniques from the audio engineering community.
The incremental improvement in frequency response fidelity, as
well as the incremental cost in terms of gate count, is presented
for each of these techniques. The complete Verilog-coded
implementation requires low tens of thousands of logic gates
and 10 kB of memory. Experimental results are provided for
four different PWM dc–dc converters, including a synchronous
buck, CCM boost, and DCM boost, showing the fidelity of the
results that can be obtained. In addition, waveforms of output
voltage during the identification process show that the identification can be accomplished in several hundred milliseconds
and that the output voltage can be kept within specified bounds
during the entire process.
Although the results shown here are limited to first- and
second-order switching converters, the cross correlation approach as well as the fractional-bandwidth smoothing and
impulse-response truncation techniques are directly applicable
to higher order systems, e.g., converters with input filters or
resonant converters. However, for optimum results, the preemphasis/postemphasis filter design may need to be modified for
converters with very large dynamic range, very steep magnitude
roll-off, and/or valleys in the magnitude response.
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2534
IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 23, NO. 5, SEPTEMBER 2008
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Mariko Shirazi received the B.S. degree in mechanical engineering from the University of Alaska Fairbanks in 1996, and the M.S. degree in electrical engineering in 2007 from the University of Colorado,
Boulder, where she is currently working toward the
Ph.D. degree in electrical engineering.
From 1996 to 2004, she was an Engineer at
the National Wind Technology Center, National
Renewable Energy Laboratory, where she was
engaged in research on the design and deployment of
hybrid wind–diesel power systems for village power
applications. Her current research interests include system identification and
autotuning of digitally controlled switched-mode power supplies.
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SHIRAZI et al.: FREQUENCY RESPONSE MEASUREMENT CAPABILITIES IN DIGITAL CONTROLLERS FOR DC–DC CONVERTERS
Jeffrey Morroni (S’06) received the B.S. and
M.S. degrees in electrical engineering in 2008 from
the University of Colorado, Boulder, where he is
currently working toward the Ph.D. degree in power
electronics.
His current research interests include adaptive
tuning and control of power electronics systems.
Arseny Dolgov received the B.S. degree in aerospace
engineering sciences in 2007 from the University of
Colorado, Boulder, where he is currently working
toward the M.S. degree in electrical engineering,
and also, working at the Colorado Power Electronics
Center.
His current research interests include low-power
wireless sensors and RF energy harvesting.
Mr. Dolgov was awarded the Dean's Outstanding
Graduate for Research Award by the University of
Colorado.
2535
Regan Zane (SM’07) received the B.S., M.S., and
Ph.D. degrees in electrical engineering from the
University of Colorado, Boulder, in 1996, 1998, and
1999, respectively.
In 2001, he joined the University of Colorado as
a Faculty Member, where he is currently an Associate Professor of Electrical Engineering. During
1999–2001, he was with the GE Global Research
Center, Niskayuna, NY, where he developed custom
integrated circuit controllers for power management
in electronic ballasts and lighting systems. His
current research interests include energy-efficient lighting systems, adaptive
and robust power management systems, and low-power energy harvesting for
wireless sensors.
Dr. Zane was the recipient of the National Science Foundation (NSF)
CAREER Award in 2004 for his work in energy efficient lighting systems,
the 2005 IEEE Microwave Best Paper Prize, the University of Colorado 2006
Inventor of the Year Award and the 2006 Provost Faculty Achievement Award,
and the 2008 John and Mercedes Peebles Innovation in Teaching Award.
He is currently an Associate Editor of the IEEE TRANSACTIONS ON POWER
ELECTRONICS LETTERS, and a member-at-large of the IEEE Power Electronics
Society AdCom.
Dragan Maksimovic (SM’05) received the B.S. and
M.S. degrees in electrical engineering from the University of Belgrade, Yugoslavia, in 1984 and 1986,
respectively, and the Ph.D. degree from the California
Institute of Technology, Pasadena, in 1989.
From 1989 to 1992, he was with the University of
Belgrade. Since 1992, he has been with the Department of Electrical and Computer Engineering, University of Colorado, Boulder, where he is currently
a Professor and the Director of the Colorado Power
Electronics Center (CoPEC). His current research interests include digital control techniques and mixed-signal integrated circuit design for power electronics.
Prof. Maksimovic was the recipient of the National Science Foundation
CAREER Award in 1997, the Power Electronics Society TRANSACTIONS Prize
Paper Award in 1997, the Bruce Holland Excellence in Teaching Award in
2004, and the University of Colorado Inventor of the Year Award in 2006.
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