2524 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 0885-8993/$25.00 © 2008 IEEE Authorized licensed use limited to: UNIVERSITY OF COLORADO. Downloaded on December 3, 2008 at 15:41 from IEEE Xplore. Restrictions apply. 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, 2525 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 Authorized licensed use limited to: UNIVERSITY OF COLORADO. Downloaded on December 3, 2008 at 15:41 from IEEE Xplore. Restrictions apply. 2526 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 Authorized licensed use limited to: UNIVERSITY OF COLORADO. Downloaded on December 3, 2008 at 15:41 from IEEE Xplore. Restrictions apply. 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 2527 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. Authorized licensed use limited to: UNIVERSITY OF COLORADO. Downloaded on December 3, 2008 at 15:41 from IEEE Xplore. Restrictions apply. 2528 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. Authorized licensed use limited to: UNIVERSITY OF COLORADO. Downloaded on December 3, 2008 at 15:41 from IEEE Xplore. Restrictions apply. SHIRAZI et al.: FREQUENCY RESPONSE MEASUREMENT CAPABILITIES IN DIGITAL CONTROLLERS FOR DC–DC CONVERTERS 2529 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 Authorized licensed use limited to: UNIVERSITY OF COLORADO. Downloaded on December 3, 2008 at 15:41 from IEEE Xplore. Restrictions apply. 2530 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 23, NO. 5, SEPTEMBER 2008 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, Authorized licensed use limited to: UNIVERSITY OF COLORADO. Downloaded on December 3, 2008 at 15:41 from IEEE Xplore. Restrictions apply. SHIRAZI et al.: FREQUENCY RESPONSE MEASUREMENT CAPABILITIES IN DIGITAL CONTROLLERS FOR DC–DC CONVERTERS 2531 = 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 =26 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 Authorized licensed use limited to: UNIVERSITY OF COLORADO. Downloaded on December 3, 2008 at 15:41 from IEEE Xplore. Restrictions apply. 2532 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 23, NO. 5, SEPTEMBER 2008 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 Authorized licensed use limited to: UNIVERSITY OF COLORADO. Downloaded on December 3, 2008 at 15:41 from IEEE Xplore. Restrictions apply. 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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B. Pearson, “optimal-feedback controllers for MIMO discrete-time systems,” IEEE Trans. Autom. Control, vol. 32, no. 4, pp. 314–322, Apr. 1987. 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. Authorized licensed use limited to: UNIVERSITY OF COLORADO. Downloaded on December 3, 2008 at 15:41 from IEEE Xplore. Restrictions apply. 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. Authorized licensed use limited to: UNIVERSITY OF COLORADO. Downloaded on December 3, 2008 at 15:41 from IEEE Xplore. Restrictions apply.

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