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Discrete signal processing techniques for power converters : multi-carrier modulation and efficient filtering… Forbes, Jason 2013

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Discrete Signal ProcessingTechniques for PowerConvertersMulti-Carrier Modulation and EfficientFiltering TechniquesbyJason ForbesB.Eng., Memorial University of Newfoundland, 2011A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Electrical and Computer Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)October 2013? Jason Forbes 2013AbstractDigital control has become ubiquitous in the field of power electronics due to the ease ofimplementation, reusability, and flexibility. Practical engineers have been hesitant to usedigital control rather than the more traditional analog control methods due to the unfamiliartheory, relatively complicated implementation and various challenges associated with digitalquantization. This thesis presents discrete signal processing theory to solve issues in digitallycontrolled power converters including reference generation and filtering.First, this thesis presents advancements made in the field of digital control of dc-ac andac-dc power converters. First, a multi-carrier PWM strategy is proposed for the accurate andcomputationally inexpensive generation of sinusoidal signals. This method aims to reducethe cost of implementing a sine-wave generator by reducing both memory and computationalrequirements. The technique, backed by theoretical and experimental evidence, is simple toimplement, and does not rely on any specialized hardware. The method was simulated andexperimentally implemented in a voltage-controlled PWM inverter and can be extended toany application involving the digital generation of periodic signals.The second advancement described in this thesis is the use of simple digital filters toimprove the response time of single-phase active rectifiers. Under traditional analog controlstrategies, the bandwidth of an active rectifier is unduly restricted in order to reduce anyunwanted harmonic distortion. This work investigates digital filters as a proposed means toimprove the bandwidth, and thereby create a faster, more efficient ac-dc power converter.Finally, a moving average filter is proposed, due to its simple implementation and minoriiAbstractcomputational burden, as an efficient means to expand the bandwidth. Since moving averagefilters are well known and widely understood in industry, this proposed filter is an attractivesolution for practicing engineers.The theory developed in this thesis is verified through simulations and experiments.iiiPrefaceThis work is based on research performed at the Electrical and Computer Engineering de-partment of the University of British Columbia by Jason Forbes, under the supervision of Dr.Martin Ordonez. Some experimental validation work was done in collaboration with MatiasAnun.A version of Chapter 2 has been submitted for publication [1].Versions of Chapter 3 and Chapter 4 have been published at the IEEE Applied PowerElectronics Conference and Exposition (APEC) and IEEE Energy Conversion Congress andExposition (ECCE), respectively [2],[3].As first author of the above-mentioned publications, the author of this thesis developed thetheoretical concepts, performed simulation, and wrote the manuscripts. Advice and technicalsupport was provided from Dr. Martin Ordonez in all three papers. MASc. student MatiasAnun developed the experimental platforms and aided in producing and summarizing theexperimental captures for papers [2],[3] .ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2.1 Digital Sinusoidal Synthesis . . . . . . . . . . . . . . . . . . . . . . . 71.2.2 Voltage Source Inverters . . . . . . . . . . . . . . . . . . . . . . . . . 91.2.3 PFC Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Derivation and Analysis of Multi-Carrier PWM . . . . . . . . . . . . . . . 162.1 Harmonic Spectrum of Multi-Carrier Digital Synthesis . . . . . . . . . . . . 16vTable of Contents2.2 Harmonic Distortion of Multi-Carrier Based Sinusoidal Reference . . . . . . 232.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 Implementation of Multi-Carrier PWM for Inverter Applications . . . . 283.1 Period Resolution of Multi-Carrier Digital Synthesis . . . . . . . . . . . . . 283.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.3 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 Ripple Elimination In Closed-Loop Digital Converters . . . . . . . . . . . 424.1 Analysis of a Moving Average Filter . . . . . . . . . . . . . . . . . . . . . . 424.2 Simulation and Experimental Results . . . . . . . . . . . . . . . . . . . . . 474.3 Filter Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55viList of Tables3.1 Multi-carrier 8-bit frequency variation patterns. . . . . . . . . . . . . . . . . 333.2 THD simulation results for sinusoidal generation. . . . . . . . . . . . . . . . 383.3 Component list and design parameters for inverter prototype. . . . . . . . . 404.1 Component list and design parameters for PFC. . . . . . . . . . . . . . . . . 494.2 Comparison of digital filter techniques. . . . . . . . . . . . . . . . . . . . . . 51viiList of Figures1.1 Conceptual operation of non-uniformly sampled sinusoidal synthesis. Note thedistortion that exists due to the nonuniform quantization error. . . . . . . . 21.2 Uniformly sampled sinusoidal synthesis exhibiting frequency resolution. . . 51.3 Circuit diagram of a PFC boost converter with a voltage feed-forward currentloop and placement of MAF filter in the voltage feedback loop. . . . . . . . 61.4 Without MAF in the feedback loop, output voltage experiences a recoverytime of seven line cycles and inductor current has a THD of 3%. . . . . . . . 71.5 (a) Time-domain output of a single-phase PFC exhibiting a ripple with fun-damental frequency of twice the line frequency. (b) Harmonic spectrum of aPFC dc-output with inverter-generated input signal, visualized using a 600ptHamming window. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1 Reference generation procedure for digital PWM using traditional look-up ta-ble of sizes N = 64 (top) and N = 1024 (middle) compared with the proposedmulti-carrier method (bottom). The proposed method uses less memory, elim-inates sub-harmonic oscillations, and achieves less THD. In this figure, thetime-domain distortion is exaggerated for emphasis. . . . . . . . . . . . . . . 172.2 Reference generated by outputting N = 8 samples using multi-carrier sinu-soidal synthesis (top). The samples, stored in a look-up table, are generatedby uniformly sampling an ideal waveform (bottom). . . . . . . . . . . . . . . 18viiiList of Figures2.3 The frequency spectrum of a multi-carrier sinusoidal reference generated froma table of N = 12 points. The spectrum is periodic, with period 1/THz. . . 232.4 Total harmonic distortion of traditional non-uniformly sampled reference gen-eration and multi-carrier reference generation. The THD decreases as the term?T/T decreases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.1 Frequency resolution capabilities in uniform, non-uniform, and multi-carriersinusoidal synthesis, presented for a microcontroller with fclk = 0.6MHz,N = 120 and variable fundamental frequencies. Uniformly-sampled refer-ence has frequency quantization error that grows with both fundamental andcarrier frequency. Multi-carrier frequency error only grows with fundamen-tal frequency, and non-uniform carrier generation has frequency quantizationerror that grows with carrier frequency. . . . . . . . . . . . . . . . . . . . . . 313.2 A functional diagram of a multi-carrier based single-phase inverter. . . . . . 323.3 Output waveforms of the proposed multi-carrier method. Example multi-carrier patterns of ratio ?/N are used to minimize the THD of the modulatedsine wave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.4 Multi-carrier PWM algorithm. After the fundamental period is determined,this algorithm is used to switch between two carriers to synthesize the referencefrequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.5 Simulation results for a look-up table generated reference (top) and invertercapacitor voltage (bottom) for a reference generated with a look-up table ofsize N = 64 and non-uniform sampling with r = 1.0016. (a) Results presentedin mV, relative to the fundamental harmonic to show relative strength ofsubharmonic distortion (b) Entire spectrum presented in dB, relative to thefundamental harmonic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36ixList of Figures3.6 Simulation results for a look-up table generated reference (top) and invertercapacitor voltage (bottom) for a reference generated with a look-up table of sizeN = 1024 and non-uniform sampling with r = 16.0256. (a) Results presentedin mV, relative to the fundamental harmonic to show relative strength ofsubharmonic distortion (b) Entire spectrum presented in dB, relative to thefundamental harmonic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.7 Simulation results for a look-up table generated reference (top) and invertercapacitor voltage (bottom). A multi-carrier look-up table of size N = 64 with?/N = 32/64. (a) Results presented in mV, relative to the fundamental har-monic to show relative strength of subharmonic distortion (b) Entire spectrumpresented in dB, relative to the fundamental harmonic. . . . . . . . . . . . . 383.8 The full-bridge inverter prototype used in experiments comprised of the controlcard (left) and invterter module (right). . . . . . . . . . . . . . . . . . . . . 393.9 The output spectrum of an inverter implementing a look-up table of size N =64. (a) Non-uniform sampling with r = 1.0016 to ensure an output frequencyof 50.0Hz, (b) A multi-carrier look-up table with ?/N = 32/64 to ensure anoutput frequency of 50.0Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.1 The signal diagram of a MAF filter in a fixed-point microprocessor showingQ-notation and implicit division. . . . . . . . . . . . . . . . . . . . . . . . . 444.2 Simulation results for a PI compensated PFC with and without a MAF filter.(a) The capacitor voltage transient after a current load step-up. (b) Theinductor current transient current load step-up. . . . . . . . . . . . . . . . . 484.3 The single phase PFC prototype used in experiments comprised of the controlcard (left) and PFC (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . 49xList of Figures4.4 Experimental results for a PI compensated PFC for current step: (a) Withouta MAF filter, (b) With a MAF filter of kernel size M + 1 = 64. For bothcaptures: input voltage (Ch1), input current (Ch2), output voltage (Ch3).For (b) only: MAF-averaged output voltage (Ch4). . . . . . . . . . . . . . . 504.5 Amplitude and phase response comparison of a MAF, Comb and digital notchfilter with a roll-off factor of r = 0.825. (b)The signal diagram of a MAF filtershowing Q-notation and implicit division. . . . . . . . . . . . . . . . . . . . 51xiAcknowledgmentsI would like to acknowledge my senior supervisor, Dr. Martin Ordonez, not only for histechnical guidance, but also for his support and encouragement throughout the past twoyears. He helped evolve my technical grasp of the subjects presented within, and reinforcedmy natural curiosity about the applied sciences, helping me to become a better researcher.For this and more I wish to thank him.I would also like to thank the other research members of the Alpha Technologies PowerLaboratory at the University of British Columbia?s Vancouver campus. In particular, Iwould like to thank Matias Anun, who developed invaluable experimental setups, and IgnacioGaliano Zubrugen, for indulging my occasional need for a sounding board.Lastly, I would like to acknowledge the National Science and Engineering Research Councilfor their support and funding.xiiFor my parentsxiiiChapter 1Introduction1.1 MotivationMicroprocessors and digital controllers are becoming more common in power converters dueto their increased flexibility, robustness, and reproducibility. The proliferation of digitallycontrolled power converters allows for numerous advancements over a wide range of top-ics. This work examines two such advancements. First, it examines advancements made tothe digital generation of sinusoidal signals used as reference functions in digital pulse-widthmodulation (PWM). The proposed advancement is named multi-carrier PWM and uses twointerleaved carrier frequencies to balance distortion and frequency resolution in referencewaveforms without sacrificing computational speed. This is a necessary component for thecontrol of many power-converters relying on a digitally generated periodic reference, includ-ing inverters and power factor correctors (PFC). The second contribution made in the areaof discrete signal processing is applying the mature field of digital filters to the problem ofPFC voltage feedback control. Through this approach, a more efficient, cost-effective PFCcan be developed with a fast transient response and low computational overhead. Althoughoften overlooked, simple design and theory are important considerations for the developmentof power converters for practical applications.Digital generation of sinusoidal signals is a necessary aspect of modern digital systems.Due to the transcendental nature of the sinusoid, there is no simple formula that can be usedto calculate the function sin(?) for an arbitrary input ?. Instead, as microprocessors become11.1. Motivationmore powerful, various techniques (including interpolation and expansive look-up tables)have become the preferred approach for digital sinusoidal synthesis, but the affordabilityof these systems relies on the existing need for external memory and powerful processors.In low-margin embedded control applications, such as digitally-controlled power converters,computation and memory is at a premium. For sub-dollar microcontrollers, excessive com-putation time can detract from the efficient control of the converter. Similarly, requiringexternal memory for an expansive look-up table can unnecessarily increase the cost, size, andoverhead of the power converter.T Tg0g0g1g2g2g3g3g4g5g5r = 1.5? look-up table output Figure 1.1: Conceptual operation of non-uniformly sampled sinusoidal synthesis. Note thedistortion that exists due to the nonuniform quantization error.Existing sinusoidal synthesis techniques generate sine signals through a look-up table[4]. Using one approach called non-uniform sampling, look-up table based sinusoidal synthe-sis generates harmonic distortion that is highly dependent on microprocessor word-length[5],[6]. This approach consists of approximating a single period of sin(?) with N sam-ples. The N samples correspond to N equidistantly spaced angles, 0 <= ?k <= 2pi wherek ? 0, 1, 2, ..., N ? 1 (due to symmetry, in practice only N/4 points are actually stored in21.1. Motivationmemory). The approximation occurs due to a non-linear quantization effect seen in Fig. 1.1.A desired angle ?? is approximated by the closest stored angle ?k, such that the fundamentalperiod T0 is precisely chosen. Operating in this fashion, the digital output of the referenceis given by the following equation, where T is the carrier period and r is a coefficient whichadjusts the sampling rate, subject to 0 < r ? N2 . In the following equations, < x >y is themodulo operation on x and y, and round(?) is an operation that returns the nearest integer.x[k] = sin(?round(rk)?N2piNT)(1.1)It can be seen from (1.1) that the fundamental period is given by the following equation.T0 =NTr(1.2)Using the notation provided above, the desired angles are determined to be ??k = rk 2piN .The angles, which can be approximated to machine precision, are as follows.? ={k2piN| 0 < k < N, k ? N}(1.3)For a given angle ??k, the angle quantization error, typically modeled as uniform quanti-zation error, can therefore be defined as:??k = inf{|??k ? ?| | ?+ 2pin ? ?, n ? I}= |round(rk)? rk| 2piN(1.4)The quantization error of the output signal is complicated by its nonuniform nature.31.1. Motivation?xk = 2sin(??k2)cos((round(rk) + rk) piN)? 2sin(??k2)cos(rk2piN)(1.5)Equation (1.5) shows that the output error is dependent on both the angle being gen-erated, and a nonlinear function of the quantization error in that angle. As will be seen,traditional methods of minimizing 1.5 rely on either increasing N or on interpolation. Anexample of the distortion generated by non-uniformly sampling the look-up table is shownin Fig. 1.1.The multi-carrier sinusoidal synthesis approach presented in this thesis completely elimi-nates the sub-harmonics introduced by non-uniform sampling, and reduces the total harmonicdistortion (THD). The interleaving of two distinct carrier frequencies as a means of eliminat-ing harmonics is analogous to the current trend of employing distinct dc-sources in multi-levelinverters for THD reduction [7], [8].The second approach, called uniform sampling, also utilizes a look-up table. In thisapproach, the coefficient r is limited to an integer, such that ??k ? ?. When this approachis used, there is no quantization error in the output apart from machine precision, sinceround(rk) = rk for r, k ? I. This approach is visualized in Fig. 1.2. While effective ineliminating the quantization error described in (1.5), uniformly-sampled sinusoidal generationunnecessarily limits the set of frequencies that can be synthesized, as will be shown in Chapter3. Both methods described above rely on the accuracy of the assumption that the carrierfrequency T is fixed. This work is motivated by the need for an efficient, computationallyinexpensive method for sinusoidal synthesis that balances quantization error and memorysize without limiting the set of achievable frequencies.One of the many applications for sinusoidal synthesis is for reference generation in pulse-41.1. MotivationT1T2T2N?TT2T1T1g0g1g2g0g1g2Figure 1.2: Uniformly sampled sinusoidal synthesis exhibiting frequency resolution.width modulated (PWM) controlled power converters. An inexpensive approach to eliminat-ing subharmonic distortion and decreasing THD in digital reference generation will allow formore efficient power converters, implemented using low-cost hardware. This work will specif-ically study carrier-based single-phase inverters as an application for the sinusoidal synthesistechniques discussed within.The second contribution to digital signal processing for power converters in this thesisis the elimination of harmonic content from the control loop of active rectifiers, also knownas power factor correctors (PFC). A PFC is an ac-dc converter that aims to match theinput voltage and current in both shape and phase [9]. The circuit diagram of a PFC withtraditional feedback loops is shown in Fig. 1.3. The input voltage for the PFC is a sinusoidalsignal, typically at line frequency. The PFC attempts to emulate a resistive load by shapingthe inductor current iL to match the input voltage |vi|. To accomplish this, a variety ofdifferent current controllers are used [9]-[11]. The inner control loop compares the inductorcurrent to a current reference to determine the switching pattern necessary to achieve the51.1. Motivationcorrect waveform. An outer control loop is needed to set the control voltage v?control, whichcontrols the dc output voltage.?CioCDSLVrefMAFGcvGciPWMiLv iv ivov control^Figure 1.3: Circuit diagram of a PFC boost converter with a voltage feed-forward currentloop and placement of MAF filter in the voltage feedback loop.Typically, to facilitate the rectification of a sinusoidal inductor current, the output ca-pacitor voltage will need to exhibit a low frequency ripple, the size of which is determinedby the size of the capacitor [12]. If the spectrum of the bus voltage is examined, it will befound to contain components at the ripple frequency (which is twice the line frequency) andits harmonics. The reason for which will be examined as part of the literature review. Whenthese components are fed back through the current controller, they can cause an unwantedincrease in inductor current THD. Ideally, the signal of the bus voltage will contain no rippleor harmonics that influence the current control loop.The traditional method of reducing the harmonic content relies on techniques developedfor analog systems, the most common of which is a PI controller. The PI controller and plantact as a low-pass filter, which reduces ac content in the output signal such that the inductorcurrent THD is minimized. This action unduly restricts the bandwidth, and therefore thetransient response, of the PFC, as can be seen in Fig. Literature ReviewVrTHD = 3%tloadtrecoveryvoiLFigure 1.4: Without MAF in the feedback loop, output voltage experiences a recovery timeof seven line cycles and inductor current has a THD of 3%.1.2 Literature Review1.2.1 Digital Sinusoidal SynthesisFrom (1.1), it is readily seen that for an integer r, the function f [n] returns a uniformlysampled sine wave but, as will be shown, this constraint can limit the period resolution. Ithas been shown that for non-integer r, there is adequate resolution. For an r representedby an unsigned integer of size n-bits, the frequency resolution ?f0 = 1T2?n , where T is thecarrier period. Using a 16? bit application, this resolution is adequate for many applications[13].Using this method, the resulting non-uniformly sampled function introduces distortion. Ithas been shown that, generally, the THD introduced by this method is dependent on the valueof r. More specifically, the distortion depends upon theM whereM = inf {M |Mr ? N,M ? N}[14]. Furthermore, in the case of sinusoidal oscillators, the maximum and minimum THD is71.2. Literature Reviewgiven by M ? ? and M = 2, respectively. The fundamental power of the generated signalusing non-uniform sampling varies with the table size N , and is provided by the followingequation, assuming that the total power is Ptot = 1 and M ? ? [15].P1 =sin2(pi/N)pi/N(1.6)It has also been shown that under certain conditions, the reference waveform can exhibitsub-harmonic distortion. The condition for this is provided by the following equation [16].N < 2Mr (1.7)It can be seen from (1.6) that P1 is maximized as N ? ?. In practice, N corresponds tothe number of samples stored in a digital look-up table, and is therefore limited by the amountof available memory. Furthermore, as the sine function is expected to be called often, it isworthwhile to have the look-up table stored ?on-chip? to avoid lengthy fetch operations fromexternal memory. On-chip memory is limited for all but high-end DSPs. As an alternative,external memory is available, and commonly used in low-end microcontrollers, but this hasan associated cost that can be detrimental in low-margin industries.To circumvent the need for an excessively large look-up table, an alternative method, andone that is used in practice, is to interpolate between look-up table entries in order to reduceerror in non-integer sample rates. One of the most common techniques, and one that is oftenused in practice, is Taylor expansion around the desired point sin(??) where h is the differencebetween the desired angle ?? and the nearest stored angle ?k [17].sin(??) = sin(?k) + h cos(?k) +O(h2) (1.8)If N , the length of the look-up table, is a multiple of four, a phase shift of pi/2 corresponds81.2. Literature Reviewexactly to a N/4 samples such that cos(?k) = sin(?k+N/4), (1.8) can be rewritten as follows.sin(??) = sin(?k) + h sin(?k+N/4) +O(h2) (1.9)This form of interpolation relies on two fetch operation, one multiplication, and oneaddition; moreover, it carries the computational burden required in determining h and thenearest stored value ?k. In modern microcontroller libraries, a call to a similar interpolatinglook-up table takes 40 clock-cycles [18]. The multiplication has been found to be a significantcontributor to increased computational burden, and should be avoided [17].Another interpolation technique uses a polynomial interpolation by approximating a sinefunction by an 8th order polynomial. This interpolation method relies on specific hardwarefor efficient calculation, or on the use of many multipliers to perform the squaring operations.The technique exhibits low THD, but at the cost of increased computational burden [19].Quasi-linear interpolation improves upon the previous method by using a combinationof linear and parabolic polynomial interpolation techniques to generate a sine wave. Thisinterpolation provides modest gains, as the piece-wise quadratic combination is better suitedto modern parallelized and pipe-lined computer architectures. This algorithm is implementedin a specialized FPGA to take full advantage of the parallelization available. Microcontrollerand DSP implementations where hardware flexibility is not available limit the efficiency ofthis approach [20].1.2.2 Voltage Source InvertersPWM inverter systems are characterized by a high-frequency constant-amplitude pulse train(carrier). PWM is usually achieved by making a direct comparison, termed natural modula-tion, between the amplitudes of a modulating wave and a carrier wave. We cannot analyzePWM using a traditional Fourier series, since the PWM waveform is not guaranteed to be91.2. Literature Reviewperiodic. Even if it is periodic, results will not be accurate, as sideband and base-band fre-quencies will coincide. Instead a double-Fourier analysis is needed. Any fixed modulatingprocess can be modeled using a 3-dimensional model, such that the modulating waveform isrepresented as an infinitely large number of parallel contours in a 2-dimensional plane. Theheight of these contours, represented in a third dimension, correspond to the PWM output.The intersection between the contour plane and the carrier trajectory y = TT0x is projectedto produce the PWM waveform, where T is the carrier period and T0 is the fundamental pe-riod. The periodicity in both carrier and fundamental leads to the formulation of the doubleFourier series [21].This analysis is a useful tool for describing naturally-sampled PWM, and provides insighton how the harmonic spectrum of the modulated waveform is dependent on the harmonicstructure of the reference. However, this method is unable to describe the harmonic effectsthat occur in digital systems where, in contrast with regular sampling, the modulating wave-form is discretized and the pulse durations are determined by discrete modulation values.Regular sampling enables pulse configuration to be unambiguously defined, allowing for op-timum pulse positioning for the cancellation of particular harmonics, but it is dependent onuniform sampling times [22].Using these analysis tools, many control schemes and performance criteria have beendeveloped. The first performance criteria considered is the distortion factor, which is theroot-mean-square (RMS) distortion current for a given scheme normalized against the RMSdistortion current based on a six-step inverter, which removes the influence of the machinedependent inductance. The next performance criterion is the harmonic spectra, which canbe calculated using the analysis tools mentioned above, and from which the THD can becalculated. This is the analysis criterion of choice. Other criteria include torque ripple,switching frequency, and control-loop bandwidth [24]. The THD equation that will be usedas the primary criterion for analysis is presented below, where Pi is the power of the ith101.2. Literature Reviewharmonic.THD =???i=2 P 2iP1(1.10)Improving the harmonic content of carrier-based PWM converters is important as low-harmonic and sub-harmonic interference can affect performance in applications such as grid-connected PV inverters [25]. Multi-carrier PWM is one in a series of novel PWM tech-niques developed recently to achieve lower THD in inverters, including carrier phase shiftingin parallel-connected inverters [26], carrier phase-shifting in cascaded inverters [27], level-shifting in multi-level inverters [28], and decomposing the reference in multiple dimensionsfor two-level converters [29]. Recent work has shown that there is still significant reductionsto be made in THD through modulation strategies based on non-uniform sampling. Onemethod implements a nonuniform sampling function that can be reconstructed using a dualwavelet function [30], [31].1.2.3 PFC ControlAs described above, the dc-voltage output will contain a ripple twice the frequency of theinput sinusoidal signal. This is further complicated by the existence of higher-order harmonicsin the input signal due to rectification; these harmonics will also exist in the output voltagefor single-phase PFCs [32]. The time domain output of a single-phase PFC with a rippleand harmonics is shown in Fig. 1.5(a). As can be seen in the frequency domain, shown inFig.1.5(b), this ripple is comprised of frequency components at twice the line frequency andits harmonics. The feedback of this harmonic content will create an unwanted distortion inthe PWM control signal.There are many control techniques for the current-shaping in PFCs. Three-phase systemscan take advantage of space-vector and dead-beat control strategies [33]. Other methods use111.2. Literature Review200210220230240 [V]T0 line 2Tline~ v      busTime (s)(a)?50050100150  |V(f) [dB]|2fFrequency (Hz)l 4f l 6f l(b)Figure 1.5: (a) Time-domain output of a single-phase PFC exhibiting a ripple with funda-mental frequency of twice the line frequency. (b) Harmonic spectrum of a PFC dc-outputwith inverter-generated input signal, visualized using a 600pt Hamming window.a zero-cross detector coupled with an input-current estimator to determine the PFC switchingpattern [34] and other predictive control methods [35],[36]. The current-controller chosen inthis work is the popular average-current control, seen in Fig. 1.3 [9]. In this control scheme,the dc output-voltage of the single-phase PFC is used to scale the input voltage, the resultof which is used as a reference for the current-controller. If there is any harmonic contentin the dc output-voltage, the current reference will be distorted, and therefore the inductorcurrent will exhibit distortion.Using small signal analysis, the voltage loop of a PFC can be modeled as a simplified singlepole [9]. With a PI controller, the voltage loop of a PFC can be controlled to produce anarbitrary large bandwidth, only limited by the sample rate. But with such a large bandwidth,the ripple and harmonics in the output voltage will become amplified and greatly distort theinductor current. To reduce this, the PI compensation is adjusted to produce an attenuatingeffect for signals at and above the ripple frequency. It has been shown that to provideadequate attenuation, the system bandwidth is greatly reduced, often to values less than1Hz [10]. With such a low bandwidth, the traditionally-compensated PFC is unable to121.2. Literature Reviewquickly respond to transients.Previous digital filters have been proposed to overcome this bandwidth limitations byusing discrete algorithms. A two-pole, two-zero digital notch filter has been proposed andimplemented as a simple means to remove the ripple from the voltage feedback loop. Thisdigital notch filter is capable of filtering out the ripple frequency, but the harmonics remain[37]. The transfer function for the digital notch filter is given by the following equation,where f0 is the filter frequency, fs is the sample frequency, and r is a parameter between0 < r < 1 which determines the steepness of the filter roll-off. This filter has the advantageof removing the ripple frequency and requires a minimum memory storage to do so.Hz =1? 2 cos(2pifo/fs)z?1 + z?21? 2r cos(2pifo/fs))z?1 + r2z?2(1.11)It has been shown that the computation time needed for this filter can increase the controlloop by 12?s. In inexpensive micro-controllers, there is also a limitation in the precision ofthe roll-off factor r. In previously conducted experiments, the steepest roll-off factor thatcould be used without causing filter instability was r = 0.95 [38].A modified comb filter has also been presented. The comb filter is designed to notch afrequency and its harmonics while returning to a unitary gain between notches, maximizingthe bandwidth?s potential [39], [40]. The transfer function for a comb filter is given by thefollowing equation, where M is the kernel size of the filter and r is a parameter between0 < r < 1 that determine the steepness of the filter roll-off.Hz =1? z?(M+1)1? z?11? rz?11? rM+1z?(M+1)(1.12)To filter a specific frequency, f0, the kernel size M and sample frequency fs must bechosen such that fo = fs/(M + 1). From analyzing the transfer function (1.12), it can beseen that the comb filter requires 2M samples to be recorded in memory, M input samples131.3. Organizationand M output samples, for the algorithm to be computed efficiently in real-time. Similarlyto the notch filter, there is a limitation in the roll-off factor r, such that values r ? 1 increasethe chance of instability due to quantization effects.1.3 OrganizationThis thesis is organized as follows.Chapter 2 introduces the multi-carrier method as a means of decreasing harmonic dis-tortion in sinusoidal reference waveforms. This chapter derives the associated harmonicspectrum of a sinusoidal signal synthesized through such a method, after which the THDcan be compared to non-uniformly sampled sinusoidal generation. A proof is supplied whichdetermines the sufficient conditions necessary to eliminate half of the offending harmonicsand further reduce the THD. It is shown that multi-carrier PWM outperforms non-uniformreference generation in terms of THD and memory minimization.In Chapter 3, the synchronization capability for ac-dc/dc-ac power converters is consid-ered. Under digital PWM, the number of fundamental frequencies that can be achieved islimited due to quantization phenomenon inherent in digital systems. This is analyzed for uni-formly sampled and multi-carrier sinusoidal generation. It is shown that multi-carrier PWMsignificantly outperforms uniformly sampled reference generation in terms of synchronizationcapabilities. Implementation details for multi-carrier PWM in inverters are considered, andexperimental and simulation validations of the theory are presented.Chapter 4 introduces the moving average filter (MAF) as a practical and efficient digitalfiltering technology to eliminate the harmonic content in closed-loop operation of ac-dc/dc-acconverters. This filter is necessary to improve the sluggish behavior of traditionally controlledac-dc/dc-ac converters without increasing the THD. A theoretical derivation of the filter ispresented, comparing it to other common digital filtering techniques with an emphasis on141.3. Organizationminimizing computation and memory requirements. Lastly, experimental and simulationresults confirm the significant improvement in response time over PI controllers in the caseof the PFC.Lastly, Chapter 5 presents a summary of the work completed within the thesis as well asconcluding remarks.15Chapter 2Derivation and Analysis ofMulti-Carrier PWM2.1 Harmonic Spectrum of Multi-Carrier DigitalSynthesisThis chapter introduces a method of decreasing the quantization error and memory require-ments of the sine-waveform reference, while maintaining low computational burden. Thisis accomplished by interleaving two carrier frequencies, a unique technique that is ideal forPWM-based sinusoidal synthesis such as those in inverters. Each output period is composedof PWM pulses generated at two distinct carrier frequencies in order to generate a singlecoherent output waveform. Fig. 2.1 compares the advantages of the proposed method to thestandard method. Advantages include the reduction in memory size for equal THD and thecomplete elimination of sub-harmonic distortion.To evaluate the THD of the resulting signal, the frequency spectrum Y (?) is determinedin terms of the uniformly sampled signal ga(t), with frequency spectrum Ga(?). The followingdiscussion relates to the THD of the reference waveform before it is modulated by PWM,and does not include the harmonic distortion introduced by this modulation, as conventionaltechniques do not allow for variable carrier frequencies.A sequence of the samples y(t) is defined and acts as the reference for pulse width mod-162.1. Harmonic Spectrum of Multi-Carrier Digital SynthesisN = 64Memoryg0g63g0g0g6g7g11g15g17g23g24gnN = 64g0g63gnT T T TT2T1T1T0 = 0.02s?4?s N = 1024g0gng1023g0g120g241g361T T T1000T02T01T02T01T02T0Figure 2.1: Reference generation procedure for digital PWM using traditional look-uptable of sizes N = 64 (top) and N = 1024 (middle) compared with the proposed multi-carrier method (bottom). The proposed method uses less memory, eliminates sub-harmonicoscillations, and achieves less THD. In this figure, the time-domain distortion is exaggeratedfor emphasis.ulation. An example of this is shown in Fig. 2.2.y(t) =??n=??g(t)? (t? tn) (2.1)A further constraint is imposed such that only two carriers, with periods T1 and T2, ? tn ? {T1, T2} (2.2)The sequence y(t) defined in (2.1) is periodic with period T0 = NT , where T is definedas the effective carrier and N ? N is the number of samples in the period. It is clear from172.1. Harmonic Spectrum of Multi-Carrier Digital SynthesisN TN TtNt1t0TT1T2t2t3TT2Tr1 y(t)x(t)g(t3) g(t4)ga(3T)ga(4T)Figure 2.2: Reference generated by outputting N = 8 samples using multi-carrier sinusoidalsynthesis (top). The samples, stored in a look-up table, are generated by uniformly samplingan ideal waveform (bottom).this representation that the following is true.g(tn) = g(tn+N) (2.3)The sequence y(t) can be generated by transforming the uniformly sampled signal ga(t),which is also periodic in NT , as follows. Let x(t) be the signal ga(t) sampled with a samplingperiod T .x(t) =??n=??ga(t)? (t? nT ) (2.4)The sequence x(t) can be divided into N sub sequences as follows.182.1. Harmonic Spectrum of Multi-Carrier Digital Synthesisx0(t) =??n=??ga(t)? (t? nNT )x1(t) =??n=??ga(t+ T )? (t? nNT )...xm(t) =??n=??ga(t+mT )? (t? nNT )...xN?1(t) =??n=??ga(t+ (N ? 1)T )? (t? nNT ) (2.5)The Fourier transform of the mth sub-sequence can be found by applying common trans-formations [41].Xm(?) =1NT??k=??Ga(? ? k 2piNT)ej(??k2piNT )mT (2.6)The original sequence (2.1) can now be reconstructed by delaying each sub-sequence xm(t)by tm.y(t) =N?1?m=0xm(t? tm) (2.7)Due to the linearity of (2.7), its Fourier transform is easily found.Y (?) =N?1?m=0Xm(?)e?j?tm (2.8)In (2.8), Y (?) is the Fourier transform of the original sequence y(t); Xm(?) is the Fouriertransform of sequence xm(t), and Ga(?) is the Fourier transform of the uniformly sampled192.1. Harmonic Spectrum of Multi-Carrier Digital Synthesissignal which, when outputted at non-uniform times t0, t1, ..., generates the sequence y(t).Next, we define a term rm which is the phase error mT ? tm. Let us also assume that theterm Ga(?) = 2pi?(???0), as this will aid us in defining the THD of the inverter. Introducingthese terms into (2.6), a simplified expression can be developed.Y (?) = 2piT??k=??A[k]?(? ? ?0 ? k2piNT)(2.9)A[k] = 1NN?1?m=0ej?0rmejk(rm?mT )2piNT (2.10)To determine the THD of this harmonic spectrum, first we must determine the phaseerror rm. Its definition is repeated for clarity below.rm = mT ? tm (2.11)Using (2.2) and the fact that one period has a length NT , we can define an integer ?subject to the constraint 0 ? ? ? N to define the period length as follows.NT = ?T1 + (N ? ?)T2 (2.12)For the mth sample, the time tm can be determined in a similar fashion by introducingan integer ?m subject to the constraints 0 ? ?m ? m and ?N?1 = ?.tm = ?mT1 + (m? ?m)T2 (2.13)Combining (2.11), (2.12), and (2.13), the phase error can be defined in terms of ? and?m, the number of samples outputted with carrier T1 over NT and tm, respectively. We alsodefine ?T as the difference between the two carrier periods ?T = T1 ? T2.202.1. Harmonic Spectrum of Multi-Carrier Digital Synthesisrm =(m ?N? ?m)?T (2.14)To ensure the THD of the resulting output is as low as possible, the phase error isminimized with respect to ?m. How this is implemented in practice will be the subject ofChapter 3. Assuming the previously defined constraints on ?m are enforced, the followingrelationship is found.?m =?m ?N?(2.15)Substituting (2.15) into (2.14), the following relationship is found.rm = m ?N ??m ?N?= ?TN ?m?? 1?N(2.16)Using a microprocessor, the smallest difference between two carriers, ?T , is the clockperiod. For effective PWM operation, it has been proposed in [6] that 8-bits of PWMresolution is sufficient, or ?T/T < 0.004. From (2.16), it can be seen that rm ? ?T forall m ? N. It follows that rm ? ?T << T . Therefore, under typical PWM operation, itis a reasonable assumption that rm ? mT ? ?mT for all m ? N. Using this assumptionand (2.16), the spectrum coefficients A(k) for Ga(?) = 2pi?(? ? ?0) can represented by thefollowing equation.A[k] ? 1NN?1?m=0ej2pi?TN2T?m??1?N e?jkm2piN (2.17)From the frequency spectrum given in (2.9), it can be seen that the coefficients A[k]defined in (2.17) determine the amplitude of the frequency components.For a sinusoidal reference, such as that used with an inverter, the frequency spectrum of212.1. Harmonic Spectrum of Multi-Carrier Digital Synthesisthe uniformly-sampled sine wave is Ga(?) = jpi(?(? + ?0) ? ?(? ? ?0)). Using the theoryprovided above, the resulting spectrum is determined to be as follows.Y (?) =jpiT(??k=??A+[k]?(? + ?0 ? k2piNT)???k=??A?[k]?(? ? ?0 ? k2piNT))A+[k] ?1NN?1?m=0e?j2pi?TN2T?m??1?N e?jkm2piNA?[k] ?1NN?1?m=0ej2pi?TN2T?m??1?N e?jkm2piN (2.18)Employing a change of variables in the summation index and considering that A+[k] =A??[N?k], a simplified expression can be found in terms of the previously defined coefficients(2.17).Y (?) =2piT??k=??B[k]?(? ? ?0 ? k2piNT)B[k] =j2(A?[N ? k ? 2]? A[k]) (2.19)The harmonic spectrum for the proposed method is presented in Fig 2.3. The relativesize of the harmonics are shown in different scales to better show the harmonic structure.222.2. Harmonic Distortion of Multi-Carrier Based Sinusoidal Reference0T0 T0 2T11-1T1|A*[10]-A[0]||A*[8]-A[2]||A*[6]-A[4]| |A*[4]-A[6]||A*[2]-A[8]||A*[0]-A[10]|Frequency (Hz)AmplitudeFigure 2.3: The frequency spectrum of a multi-carrier sinusoidal reference generated from atable of N = 12 points. The spectrum is periodic, with period 1/THz.2.2 Harmonic Distortion of Multi-Carrier BasedSinusoidal ReferenceThe first obvious benefit of this method is that it can be designed such that the coefficientsB[k] = 0 for odd k. Sufficient conditions for this to be true are that both N and ? are even.The proof for this is as follows. For conciseness, we will introduce the constant ? = 2pi ?TN2Tand the notation < x >y to denote x mod y.232.2. Harmonic Distortion of Multi-Carrier Based Sinusoidal Reference0 =j2(A?[N ? k ? 2]? A[k])=?j2NN?1?m=0e?j2piN km(ej??m??1?N )ej2piN m?e?j??m??1?N e?j2piN m)e?j2piN m= 1NN?1?m=0e?j2piN m(k+1)sin(? ?m?? 1?N +2piNm)(2.20)Given that k is odd, k + 1 must be even; therefore we can replace it with index 2l. Wewill also use the fact that N is even to split the summation.N/2?1?m=0e?j2piN 2mlsin(? ?m?? 1?N +2piNm)= ?N?1?m=N/2e?j2piN 2mlsin(? ?m?? 1?N +2piNm)= ?N/2?1?m=0e?j2piN 2(m+N2 )lsin(??(m+ N2)?? 1?N+2piN(m+ N2))=N/2?1?m=0e?j2piN 2mlsin(??(m+ N2)?? 1?N+ 2piNm)(2.21)One case where this equality is always true is when the mth summation terms are equal.A sufficient condition for this to be true is if the arguments of both the sine functions areequal.242.2. Harmonic Distortion of Multi-Carrier Based Sinusoidal Reference? ?m?? 1?N +2piNm = ??(m+ N2)?? 1?N+ 2piNm?m?? 1?N =?(m+ N2)?? 1?N??N2??N= 0? ? is even (2.22)This proves the former claim that B[k] = 0 for odd k under the conditions that N and ?are even. This has one profound consequence; unlike other methods proposed [16],[15], thereis no possibility of subharmonic components using the multi-carrier method proposed in thisthesis.From the frequency spectrum representation of the multi-carrier based sinusoidal referencegiven in (2.19), it can be seen that the coefficients B(k) determine the amplitude of thefrequency components. Analyzing (2.17), we see that the coefficients A(k), and thereforeB(k), are periodic on k with period N . This leads to the frequency spectrum Y (?) beingperiodic on ? with period 2pi/T rads. It should also be noted that using the samplingtheorem, the unsampled signal can be reconstructed by including only harmonics of frequency|f | ? 12T [41]. Using this knowledge, the normalized total power of the reference sinusoid canbe calculated as follows.Ptotal =N/2?1?k=0|B[k]|2 (2.23)Therefore, the THD of a multi-carrier generated sinusoidal reference can be calculatedthrough the following equation.252.2. Harmonic Distortion of Multi-Carrier Based Sinusoidal ReferenceTHD =?N/2?1k=0 |B[k]|2 ? |B[0]|2|B[0]|2(2.24)Equations (1.6) and (2.24) were used to compare the THD of the traditional non-uniformsampling digital synthesis of a sinusoidal reference with that of the proposed method; theresults of this comparison are featured in Fig. 2.4. These results are based on the assumptionmade in (2.17) and are valid only for ?T  T . Due to the limits of digital PWM, in practice?T/T < 0.01, in which case the reference generated with multi-carrier PWM has a THDthat is several orders of magnitude less than that produced using traditional non-uniformsampling. If ?T is minimized, it is equivalent to the PWM clock.101 102 103 10410-1410-1210-1010-810-610-410-2100Size of look-up table NTHD  Non-uniform samplingMulti-carrier sampling?TT ?TT= 0.1?TT= 0.01?TT= 0.001Figure 2.4: Total harmonic distortion of traditional non-uniformly sampled reference genera-tion and multi-carrier reference generation. The THD decreases as the term ?T/T decreases.262.3. Summary2.3 SummaryThis chapter established the methodology for a multi-carrier sinusoidal synthesis technique forthe digital generation of a reference waveform. The multi-carrier method employs two carrierfrequencies that are varied to minimize the cumulative tracking error throughout one outputperiod. The derivation shows that for equivalently sized look-up tables, multi-carrier PWMoutperforms non-uniform sampling by greater than one order of magnitude. This methodalso eliminates all subharmonic components, which are a concern in applications includingPV-connected inverters and ac-drives. Implementation details and simulation results arepresented in the next chapter.27Chapter 3Implementation of Multi-CarrierPWM for Inverter Applications3.1 Period Resolution of Multi-Carrier DigitalSynthesisThis section will determine the period resolution of multi-carrier digital synthesis and com-pare it to the resolution of uniformly and non-uniformly sampled digital synthesis. In uniformsampling, the look-up table is sampled at an integer rate such that to retrieve the next sam-ple, a pointer is incremented a set amount. Under these circumstances, only one addition andone fetch operation is typically needed (not including the edge case), as well as a set operationfor PWM output. No multiplication is needed to generate the unitary sinusoidal reference.While such an approach does not generate any distortion in the generated reference, thediscrete set of carrier frequencies may not contain the nominal inverter frequency.For a digital PWM module, each carrier period T is an integer multiple of a clock-period.Using the notation developed earlier, the clock-period is the minimum difference between twoachievable carriers, ?T . In this case, the set of all achievable periods, ST can be determinedfrom (1.2) by defining an integer factor p, and assuming uniform sampling r = 1. The integerfactor p is set through a register in the microprocessor, and is the maximum count for thePWM counter. The set of all achievable periods is presented below.283.1. Period Resolution of Multi-Carrier Digital SynthesisST = {Np?T | p,N ? N} (3.1)The term ?T is limited by the PWM clock module, and is directly related to the cost ofthe microcontroller. The resolution of T0 can then be defined as:?T0 = inf {x? y | x, y ? ST , x 6= y}= N?T (3.2)In grid-synchronization applications, this resolution in general leads to a deviation ofthe operating inverter frequency from the grid frequency. Minimizing this deviation is animportant consideration in applications that require precises frequency synthesis. The multi-carrier method is able to greatly decrease ?T0 compared to the uniform-sampling method.For the multi-carrier method, the set of all achievable T0 is given in (3.3).ST = {?T1 + (N ? ?)T2 | ?,N ? N, ? ? N}= {(?+Np)?T | ?,N, p ? N, ? ? N} (3.3)The resolution of T0 can be defined using a similar argument as before.?T0 = inf {x? y | x, y ? ST , x 6= y}= ?T (3.4)The results of (3.4) are that the inverter frequency deviation does not grow with N , unlike293.1. Period Resolution of Multi-Carrier Digital Synthesisin the case of uniformly sampled digital synthesis. This formulation assumes that ? can takeany integer value, although as shown in Chapter 2, it is often wise to limit ? to even valuesonly. This limitation will double the quantization ?T0 in (3.4). For non-uniformly sampleddigital synthesis with r ? R, the maximum frequency deviation is related to the number ofbits used to represent r, and can be quite small, but comes with an increase in THD [16].It is often more helpful to view the frequency quantization, rather than the period quan-tization. As per the definition above, the PWM output period is composed of ? pulses atcarrier frequency 1/T1 = f1 and N ? ? pulses at carrier frequency 1/T2 = f2. Based on this,the equation for the PWM fundamental frequency can be expressed as follows.f0 =f1f2f2?+ (N ? ?)f1(3.5)The minimum change in output frequency can be determined by finding the differencebetween (3.5) and a frequency generated by incrementing ?. The result is a definition for thefrequency resolution capable of using multi-carrier PWM.?f0 =?????f0f2 ? f1f1f2f0 + (f2 ? f1)?????(3.6)It can be shown that f2? f1 = f1f2?T . Using this equation the frequency resolution canbe simplified.?f0 =f 20fclk + f0? f20fclk(3.7)In (3.7), the term fclk = 1/?T is the clock period. This equation enables an insight that303.1. Period Resolution of Multi-Carrier Digital Synthesisfor multi-carrier PWM, the frequency resolution only depends on the fundamental periodand the clock period. By performing a similar analysis for uniform sampled PWM, it can beshown that the frequency resolution for uniform sampling is ?f0 = f0Tfclk . The application ofthese insights leads to an overall quantization reduction of T0T = N .0 50 100 150 200 250 30000. Frequency (Hz)Frequency Quantization Error (Hz)  Uniformly SampledMulti-CarrierNon-uniformly SampledFigure 3.1: Frequency resolution capabilities in uniform, non-uniform, and multi-carrier sinu-soidal synthesis, presented for a microcontroller with fclk = 0.6MHz, N = 120 and variablefundamental frequencies. Uniformly-sampled reference has frequency quantization error thatgrows with both fundamental and carrier frequency. Multi-carrier frequency error only growswith fundamental frequency, and non-uniform carrier generation has frequency quantizationerror that grows with carrier frequency.Fig. 3.1 compares the frequency resolution of multi-carrier sinusoidal synthesis, uniformsampling sinusoidal synthesis, and non-uniform sampling sinusoidal synthesis of a controllerwith 16 ? bit registers. To make this comparison, a reference frequency is compared toN = 120 carrier pulses at a frequency of 1/T = 60kHz. The system chosen in this examplehas a clock frequency of fclk = 0.6MHz. It can be seen that at a reference frequencyof f0 = 60Hz, uniform sampling sinusoidal synthesis has a large quantization of 0.6Hz.By using multi-carrier sinusoidal synthesis, the frequency quantization is improved to 0.01.313.2. ImplementationAs mentioned in Chapter 1, the frequency resolution of non-uniform sampling is fixed anddependent only on the register size in the microcontroller.3.2 ImplementationTo implement multi-carrier sine generation in a microcontroller, a table of N equally spacedsamples corresponding to sin(n2piN ), n = 0, 1, ..., N ? 1 is stored in memory. Each PWMperiod, a pointer referencing the samples is incremented by a fixed amount and a new sampleis fetched. A frequency variation pattern corresponding to the ?/N states available must bestored in memory, but as this pattern is binary in nature, the total memory required is Mbits, where M is is the denominator of the reduced fraction ?/N .For a chosen average carrier period T and number of samples N , the two carriers T1, T2are chosen subject to the following constraints, where p is an integer.T1 =p?T + ?TT2 =p?T (3.8)T2 ?T ? T1Sine ReferenceCarrierLoad?CLook-Up-TablePWM_PRD0 1 0 0 1 0 1 0Frequency VariationPatternPWM Period   ?S3S4S1S2LCpFigure 3.2: A functional diagram of a multi-carrier based single-phase inverter.323.2. ImplementationEach PWM period, the PWM register PWM PRD is loaded with p + d?/N(m), whered?/N(m) is the mth entry in the frequency variation pattern for sequence ?/N and p is givenby (3.8). The register PWM PRD defines the number of clock periods ?T in one carrierperiod T . Loading the PWM PRD in this way enables switching between the two carriersT1, T2. A functional diagram of multi-carrier PWM is shown in Fig. 3.2, which describes howthe carrier period of the PWM is changed when driving an inverter. As well as the instructioncount for fetching the sample, each PWM period requires two additions, two fetch operations,and two set operation. The function d?/N(m) can be defined recursively in terms of (2.15).d?/N(m) =?m ?N?? d?/N(m? 1) (3.9)d?/N(?1) =0For a sine look-up table of size N = 16 implemented in single precision, the look-up tablehas a size of 64B. To store the frequency variation sequences for the ratios ?/N = 1/8 ? 7/8requires an additional 7B, much less than is needed to reduce the error by an order ofmagnitude using the standard method. Each 1 in the frequency variation pattern correspondsto carrier period T1, as shown in Fig. 3.3. Example 8?bit frequency variation sequences formulti-carrier sinusoidal synthesis with N = 16 is provided in Table 3.1.Table 3.1: Multi-carrier 8-bit frequency variation patterns.?2 0 1 0 0 0 0 0 04 0 1 0 0 0 1 0 06 0 1 0 1 0 0 1 08 0 1 0 1 0 1 0 110 0 1 1 0 1 1 0 112 0 1 1 1 0 1 1 114 0 1 1 1 1 1 1 1333.2. ImplementationT2T2T2 T2T1T2T1T1? PWM_PRD = p pp + d?/N(2) p + d?/N(4)PWM_PRD = p pp p + d?/N(4)N =12N =14? Figure 3.3: Output waveforms of the proposed multi-carrier method. Example multi-carrierpatterns of ratio ?/N are used to minimize the THD of the modulated sine wave.To determine which ? is needed for a given output frequency, simply rearrange (3.3). Theresult (3.10) provides a simple way of determining which frequency variation pattern to usefor optimum reference generation.? = T0?T?N?T0/?TN?(3.10)Since only integer values of ? can be implemented, it is assumed that T0/?T is an integer,which allows (3.10) to be written as (3.11). This quantization is caused by the frequencyresolution defined in (3.4).? =?round(T0?T)?N(3.11)Using the equations developed here, a simple yet extremely efficient PWM based sinu-soidal synthesis platform can be developed. It is important to note that (3.11) is very sus-ceptible to round-off error, and care should be taken to perform computations with adequateprecision. This platform allows for superior performance with modest memory requirementsand very low computational burden.The multi-carrier PWM algorithm is visualized in Fig. 3.4. The algorithm is entered at343.2. ImplementationExitModulate x[i]let i = i + 1PWM_PRD = d?/N[i%(?/N)] + pIs i ? N?YesNoCalculate ? using (3.11)let i = 0EnterGiven T0, N, rCalculate T using (1.2)Can T be generated?YesNoUse uniform samplingExitDetermine nearest p,T1, T2 | T2 < T < T1   (3.8)Figure 3.4: Multi-carrier PWM algorithm. After the fundamental period is determined, thisalgorithm is used to switch between two carriers to synthesize the reference frequency.the beginning of every output period, and the only information required before entering themulti-carrier PWM algorithm is the desired fundamental output frequency and look-up tablesize. In the case of an inverter, the fundamental frequency is the driving frequency or thegrid-connect frequency. In the case of active rectification of a sinusoidal signal, it is doublethe input frequency, which can be determined using zero-cross detection. If the fundamentalPWM output frequency is attainable using the uniform sampling method (i.e. by outputtingan integer N pulses per period at an obtainable carrier frequency) then the carrier frequencycan be set by traditional means and the algorithm is exited. When this is not the case,? needs to be calculated according to (3.11). Using this algorithm, multi-carrier sinusoidal353.3. Simulationsynthesis can be implemented in a microcontroller or DSP efficiently.3.3 Simulation0102030Frequency (Hz)20 40 60 80 100 120 140 160 1800102030Magnitude (mV)Magnitude (mV)(a)Magnitude (dB)Magnitude (dB)-300-200-1000-300-200-1000Frequency (Hz)0 1000 2000 3000(b)Figure 3.5: Simulation results for a look-up table generated reference (top) and inverter ca-pacitor voltage (bottom) for a reference generated with a look-up table of size N = 64 andnon-uniform sampling with r = 1.0016. (a) Results presented in mV, relative to the fun-damental harmonic to show relative strength of subharmonic distortion (b) Entire spectrumpresented in dB, relative to the fundamental harmonic.To verify the theory presented in this thesis, a simulation was conducted using the PLECSsimulation software. A full-bridge inverter was driven with a switching frequency of 3.19kHz.The inverter had an 100V input voltage and an LC filter with values 5.5mH and 50?F , givinga cut-off frequency of 304Hz. A 64?point look-up table was implemented, with an indexmultiplier r = 1.0016 to ensure a 50.0Hz output was generated. The amplitudes of both thereference and harmonic spectrum were scaled such that their peak amplitudes were 1V , afterwhich the harmonic spectrum of both waveforms were analyzed. These results, presented inFig. 3.5(a), show that the generated reference exhibits significant sub-harmonic distortion,which remains after the modulation and filtering. Running the simulation again, but with atable size of N = 1024 show that sub-harmonic distortion remains, but is reduced such that363.3. Simulationit is a minor contributor to the total harmonic spectrum, as seen Fig. 3.6(a). All simulationsused 10Hz frequency divisions to represent harmonic spectrum including sub-harmonic andovertone effects.01230123Magnitude (mV)Magnitude (mV)Frequency (Hz)20 40 60 80 100 120 140 160 180(a)Magnitude (dB)Magnitude (dB)-300-200-1000-300-200-1000Frequency (Hz)0 1000 2000 3000(b)Figure 3.6: Simulation results for a look-up table generated reference (top) and inverter ca-pacitor voltage (bottom) for a reference generated with a look-up table of size N = 1024 andnon-uniform sampling with r = 16.0256. (a) Results presented in mV, relative to the fun-damental harmonic to show relative strength of subharmonic distortion (b) Entire spectrumpresented in dB, relative to the fundamental harmonic.Next, the proposed multi-carrier method was implemented by switching between twocarrier frequencies 1/T1 = 3.19kHz and 1/T2 = 3.21kHz. The frequency variation ratio isdetermined through (3.11) to be ?/N = 32/64, and is chosen to ensure an output frequencyof 1/T = 50.0Hz. The results, presented in Fig. 3.7(a), show that the generated referenceexhibits no sub-harmonic distortion.Using the simulation data, the THD was calculated and presented in Table 3.2. It canbe seen that for the reference signal, the multi-carrier sinusoid has a much lower THD thando non-uniform sinusoidal synthesis, as predicted in Fig.2.4. This difference is largely miti-gated by increasing the table size to N = 1024 for non-uniformly sampled sinusoidal genera-tion. When comparing the THD of the inverter?s capacitor voltage, it can be seen that thisreference-induced harmonic distortion leads to a 2% change, and does so without increasing373.4. Experimental Results01230123Magnitude (mV)Magnitude (mV)Frequency (Hz)20 40 60 80 100 120 140 160 180(a)Magnitude (dB)Magnitude (dB)-300-200-1000-300-200-1000Frequency (Hz)0 1000 2000 3000(b)Figure 3.7: Simulation results for a look-up table generated reference (top) and invertercapacitor voltage (bottom). A multi-carrier look-up table of size N = 64 with ?/N = 32/64.(a) Results presented in mV, relative to the fundamental harmonic to show relative strengthof subharmonic distortion (b) Entire spectrum presented in dB, relative to the fundamentalharmonic.Table 3.2: THD simulation results for sinusoidal generation.Reference Capacitor VoltageWaveform WaveformMulti-Carrier7.85? 10?3 % 2.77%N = 64Non-uniform3.58% 4.81%N = 64Non-uniform1.69? 10?1 % 2.83%N = 1024the number of points N or using computationally expensive interpolation.3.4 Experimental ResultsTo verify the multi-carrier PWM technique, the algorithm was tested in a Texas InstrumentsTMS320F28335 DSP. A 100V input voltage was applied to a full-bridge inverter, which is383.4. Experimental ResultsFigure 3.8: The full-bridge inverter prototype used in experiments comprised of the controlcard (left) and invterter module (right).Nonuniform sampling50 Hz 100 Hz 150 Hz(a)Multi-carrier generation50 Hz 100 Hz 150 Hz(b)Figure 3.9: The output spectrum of an inverter implementing a look-up table of size N = 64.(a) Non-uniform sampling with r = 1.0016 to ensure an output frequency of 50.0Hz, (b) Amulti-carrier look-up table with ?/N = 32/64 to ensure an output frequency of 50.0Hz393.4. Experimental ResultsTable 3.3: Component list and design parameters for inverter prototype.Component Part-Number?C Texas Inst. TMS320F28335MOSFET S1,S2,S3,S4 Micrel Inc. MIC4421ZM TRC 50 ?FL 3 mHDesign Parameter Valuevin 100 VT0 0.02 sN 641/T1 3.19 kHz1/T2 3.21 kHz1/Tclk 1 MHzm, modulation index 0.9controlled by the PWM output of the microcontroller. For simplicity, a sawtooth carrierreference is used for the PWM module. The PWM output was filtered through an LC filterto produce the inverter output. The prototype and experimental set-up is shown in Fig. 3.8.Component list and design parameters are listed in Table 3.3. Refer to Fig. 3.2 for a circuitdiagram with component labels.To summarize the algorithms operating point, the experimental conditions are presentedas follows. A moderate clock of 1MHz is used to generate the DPWM output. A look-uptable of size N = 64 was uploaded to the memory to generate the sine reference. To minimizeswitching losses, a low carrier frequency of 1/T = 3.19kHz was used. Using a standardimplementation, a sampling multiplier of r = 1.0016 is necessary to accurately generate thecorrect output frequency. For multi-carrier PWM, the carrier frequencies 1/T1 = 3.19kHzand 1/T2 = 3.21kHz are used. The calculated frequency variation fraction is ?/N = 31/64,but to eliminate subharmonic components, this was changed to ?/N = 32/64.The output spectrum of the inverter caused by the signals generated through non-uniformsampling and multi-carrier PWM are presented in Fig. 3.9(a) and Fig. 3.9(b), respectively.It can be seen through inspection that there is significantly less distortion for the sinusoidal403.5. Summaryoutput generated with multi-carrier PWM, and no subharmonic distortion is visible.3.5 SummaryThis chapter established the methodology for implementing multi-carrier sinusoidal synthe-sis in low-cost microcontrollers, and by doing so greatly increases the frequency resolution ofcarrier-based ac-dc/dc-ac power converters for practical applications. Limitations of periodresolution are developed and linked to the distortion existing in the digital reference genera-tion. Simulation results verify that this distortion produces an increased THD in voltage con-trolled inverters. Experimental results were conducted, and the elimination of sub-harmoniccomponents verified. This chapter provides a simple, computationally inexpensive methodfor decreasing the THD in inverters and other sinusoidal synthesis applications without in-creasing the memory requirements nor the cost of the micro-controller.41Chapter 4Ripple Elimination In Closed-LoopDigital Converters4.1 Analysis of a Moving Average FilterTraditionally, single-phase power factor correctors suffer from a slow response time, charac-terized by a low voltage loop bandwidth (< 1Hz) that is necessary to ensure a suitable THDin the input current. This chapter examines the use of practical discrete filters to substan-tially improve the voltage feedback-loop bandwidth. As a result of the following analysis, themoving averaging filter (MAF) proves to be an effective solution. The analysis shows that itis an excellent practical filter for PFC control, and requires low computational overhead andbasic discrete signal theory literacy for its implementation.A moving average filter is defined as the average of the current input with M past samples,where M is defined as the kernel size. Mathematically, the impulse response of a MAF isdefined as:h[n] = 1M + 1M?k=0?[n? k] (4.1)where ?[n?k] is an impulse delayed by k samples. By transforming (4.1) using a discreteFourier transform (DFT) the frequency response is determined [41].424.1. Analysis of a Moving Average FilterH(ej?) = 1M + 1M?n=0e?j?n (4.2)Using the closed-form of the general geometric series, the frequency response is simplified.H(ej?) = 1M + 11? e?j?(M+1)1? e?j?(4.3)Lastly, after some simple rearrangement, an analytically descriptive representation isdeveloped.H(ej?) = 1M + 1[sin(?(M + 1)/2)sin(?/2)]e?j?M/2 (4.4)By analyzing (4.4), insights into the operation and effect of a MAF can be gleaned. Fromthe amplitude, it can be seen that a number of evenly spaced frequencies are eliminated.From the phase, it is evident that those notched frequencies correspond to a phase jump of180o. This phase jump can be interpreted in two ways. First, the frequency componentsat the notching frequencies experience an infinite time delay. In other words, the notchingfrequencies are completely removed from the system response. Secondly, the phase jump of180o signifies a point of marginal stability. Frequencies approaching the phase jump of 180oalso approach the phase-margin of the system. Components at frequencies near the notchingfrequencies may experience instabilities, and should therefore be attenuated. For the comband notch filter, this instability region can be minimized by increasing r coefficient, but thiscan increase the cost and complexity of the filter in fixed-point systems.The implementation a MAF filter in the finite impulse response (FIR) form given in (4.5)has an algorithmic asymptotic complexity of ?(M), which is not suitable for a real-timeimplementation. Instead, an infinite impulse response (IIR) form is needed to reduce thecomputational burden. The FIR form of the filter is presented below.434.1. Analysis of a Moving Average Filterx[n]z -(M+1) z[n]Figure 4.1: The signal diagram of a MAF filter in a fixed-point microprocessor showingQ-notation and implicit division.y[n] = 1M + 1M?k=0x[n? k] (4.5)To determine the IIR, the following derivation is presented. The first term can be takenout of the summation and an M + 1 term can be added to the summation as follows.y[n] = 1M + 1(M+1?k=1x[n? k] + x[n]? x[n? (M + 1)])(4.6)Next, the summation limits are adjusted by decrementing the index.y[n] = 1M + 1(M?k=0x[(n? 1)? k] + x[n]? x[n? (M + 1)])(4.7)Lastly, the summation term is recognized to be the previous output term y[n? 1].y[n] = x[n]? x[n? (M + 1)]M + 1+ y[n? 1] (4.8)The real advantage of the MAF filter is its simple implementation, even in fixed-pointmicroprocessors, as shown in Fig. 4.1, which shows the signal diagram of the algorithm andthe Q-notation representation of fixed-point signal. Notice that if M + 1 is chosen to be apower of 2, then the division is achieved by an implicit shift of the radix point. For instance,assume M + 1 = 64, therefore m = log264 = 6bits. The input value is already a fixed-point444.1. Analysis of a Moving Average Filterinteger represented by, for example, n = 8bits. Therefore, the input signal is in form Q8.0.Q-notation is the standard approach to describing fixed-point integer representation wheretwo integers represent the number of bits before and after the radix point, in this case 8and 0 respectively. After the input delay, instead of dividing the value by 64 (and losing6-bits of accuracy), the signal is assumed to be of form Q2.6. This is an implicit change inrepresentation and does not affect how the data is stored, nor does it introduce any loss ofaccuracy.Because there are no coefficients in the MAF-filter, no rounding error is introduced [42].This rounding error can be detrimental to digital filter?s performance and stability, and cangreatly influence the size of the registers associated with the algorithm [43]. Unlike the comband notch filters, the MAF filter is not affected by the coefficients? accuracy due to registersizing. Instead, the register size is only restricted by the number of bits necessary to avoidoverflow during reasonable operating conditions. The simplicity of the filter therefore leadsto a potentially cheaper and simpler implementation.For a MAF filter, the first notch of which is set to a specific frequency f0, it can be shownthat for moderate to large sized kernels, the response time is approximately the same. Toperform this analysis, we will need to define the concept of group delay. Group delay isdefined as the digital delay, measured in number of samples, that each frequency componentof an input passed through a digital filter will experience [41]. At points where the phaseresponse of the MAF filter is differentiable (i.e. everywhere besides notching frequencies),the group delay is defined as =M2(4.9)Equation (4.9) shows that, neglecting the notching frequencies, a MAF filter causes aconstant group delay of M/2. To convert the group delay to a time delay, the sample time454.1. Analysis of a Moving Average Filtermust be taken into account. The time delay caused by a MAF can be written as follows,where fs is the sample frequency of the =M2fs(4.10)The sample frequency and kernel size are constrained by the following equation, wheref0 is the frequency of the first notch.fs = (M + 1)f0 (4.11)Substituting this constraint into (4.10), the delay is determined in terms of the kernel sizeand the primary notching =M2f0(M + 1)(4.12)Minimizing (4.12), it can be seen that the fastest response is for smaller kernel sizes,M , but this can cause higher distortion in the inductor current during transients due to thesmall sample rates. As M is increased, the delay quickly approaches the maximum timedelay td = 12f0 . This means that for large kernels, the theoretically shortest recovery timefrom a step-change in the load is half the line period. This delay is larger than that seenin the comb and notch filter (of appropriately chosen r-values) and leads to a larger voltagedrop during step-transients. Fortunately, the linear phase makes it simple to compensate forthe time delay.Since the response time quickly approaches the maximum delay for large M , the sizing ofthe kernel should be chosen independently of the response time. The sample time fs shouldbe chosen such that the kernel size fs/f0 = M + 1 is a power of 2 in order to enable simplefixed-point division. Increasing M raises cost by requiring larger registers to perform the464.2. Simulation and Experimental Resultsalgorithm without introducing truncation error.4.2 Simulation and Experimental ResultsThe PFC was simulated with and without a MAF in the voltage feedback loop, as presentedin Fig. 4.2(a) and Fig. 4.2(b). The voltage and current loops are controlled with digitalPI controllers. In the simulation, a PFC is implemented with a reference output voltage ofVr = 300V , input voltage of Vi = 120Vrms, circuit parameters of L = 800?H ad C = 660?F ,and a switching frequency of fs = 60kHz. To analyze transient behavior, the system isloaded with a current step-up.Without filtering, the PI controller was calibrated to cause little overshoot while main-taining a THD of 3%. At t = 500ms the load changes from a 1A to 2A load. This causesa 5% drop in voltage and the system does not recover for t = 100ms, where recovery isjudged as a mean deviation of 1% from the reference voltage. Next, a MAF was placed inthe voltage feedback loop of the PFC. Again, a PI controller was calibrated to cause littleovershoot. The PFC was simulated with the same conditions as before, but resulted in aTHD of 1% due to the MAF. The increased bandwidth resulted in an 3% drop in the busand the system recovered in t = 25ms.To verify the MAF performance in single-phase PFCs, the algorithm was tested in aTexas Instruments TMS320F28335 DSP. A 120Vrms sinusoidal input voltage was applied to arectifier and a boost-converter was controlled to emulate a constant resistance. The prototypeand experimental set-up is shown in Fig. 4.3. Design parameters are listed in Table 4.1.Refer to Fig. 1.3 for a circuit diagram with component labels. Simulation results are verifiedexperimentally under the same operating conditions and are presented in Fig. 4.4(a) andFig. 4.4(b). The PI compensator is chosen such that the inductor current THD is 5% in bothfigures. Voltage drop and recovery time is significantly improved by the inclusion of a MAF474.2. Simulation and Experimental Results0.45 0.5 0.55 0.6 0.65 0.7285290295300305v otime (s)  With MAFWithout MAF(a)0.45 0.5 0.55 0.6 0.65 0.7012345i Ltime (s)(b)Figure 4.2: Simulation results for a PI compensated PFC with and without a MAF filter.(a) The capacitor voltage transient after a current load step-up. (b) The inductor currenttransient current load the voltage control loop, which coincides with the simulation results.484.3. Filter ComparisonFigure 4.3: The single phase PFC prototype used in experiments comprised of the controlcard (left) and PFC (right).Table 4.1: Component list and design parameters for PFC.Component Part-Number?C Texas Inst. TMS320F28335Rectifier Vishay GBPC2506MOSFET S Int. Rectifier IRFP460ADiode D Fairchild Semi. RHRP1560C 560 ?FL 577 ?HDesign Parameter Valuevi 120 Vrmsvo 300 Vfs 60 kHzMAF kernel M + 1 64iL THD 5%4.3 Filter ComparisonThe results show that by incorporating a MAF in the feedback loop, the performance of aPFC can be greatly increased. Fig. 4.5 compares the MAF with the digital notch and combfilter, for a roll-off factor of r = 0.9. The roll-off factor r determines how quickly the filterattenuates near the filter frequency. A noticeable difference in these filters are the dc gains.The MAF provides a unitary gain for the dc component, whereas the comb and notch filters494.3. Filter Comparisonviiivo28V122ms(a)viiivovoavg 12V20ms(b)Figure 4.4: Experimental results for a PI compensated PFC for current step: (a) Withouta MAF filter, (b) With a MAF filter of kernel size M + 1 = 64. For both captures: inputvoltage (Ch1), input current (Ch2), output voltage (Ch3). For (b) only: MAF-averagedoutput voltage (Ch4).will need additional gain adjustment to avoid a steady state error in the voltage controlloop. Fig. 4.5 show that all three filters reach the 180o phase shift at the same frequency,the notching frequency, giving each filter the same instability points. The larger slope, andgreater group-delay in the MAF, corresponds to a larger voltage-drop and potentially largermarginal-stability region than is present in the notch and comb filter. This is balanced bythe lack of coefficients and the resultant decrease in computation and memory requirements,effectively reducing the cost of the micro-controller. The micro-controller requirements aresummarized in Table 4.2, where computational burden is defined as the number of operations504.3. Filter Comparison0 0.1 0.2 0.3 0.4 0.500.511.52|H(?)|?/?s  MAFCombNotch0 0.1 0.2 0.3 0.4 0.5?200?1000100200<H(?)?/?sFigure 4.5: Amplitude and phase response comparison of a MAF, Comb and digital notchfilter with a roll-off factor of r = 0.825. (b)The signal diagram of a MAF filter showingQ-notation and implicit division.Table 4.2: Comparison of digital filter techniques.Memory Comp. Ripple(values) Burden EliminatedNo Filter 0 None NoNotch 4 8 OPC PartialMAF M 2 OPC YesComb 2M+2 11 OPC Yesper cycle (OPC).514.4. Summary4.4 SummaryThrough comparative analysis, a moving average filter was presented as a means to increasethe dynamic response of a typical PFC while simultaneously reducing the THD of the in-ductor current. The MAF was shown to significantly increase performance compared totraditional low-pass-filtering techniques, resulting in a PFC capable of recovering from aload-step increase in half the time obtainable traditionally. The dynamics of the MAF filterwere compared to previously proposed digital filters, such as the digital notch and combfilter. The MAF presents results comparable to the comb filter, requiring half the memoryand minimal computational burden, providing an efficient and effective solution for practicalimplementation.52Chapter 5Conclusion5.1 SummaryThis thesis introduced the multi-carrier PWM technique for digital sinusoidal synthesis. Thismethod is introduced as an alternative to non-uniformly sampled sinusoidal synthesis, whichexhibits sub-harmonic distortion and large THD, and uniformly-sampled sinusoidal synthesis,the frequency resolution of which can be unsuitable for many applications. Multi-carrierPWM has been shown to be a computationally inexpensive technique, requiring no specializedhardware, unlike previously proposed interpolation techniques.The multi-carrier technique was studied in relation to the single-phase voltage-controlledinverter. It was shown to generate an accurate reference with significantly less memory thando non-uniformly sampled sinusoidal synthesis, and simulations verified that this leads to adecrease in THD by 2%. Through experimental verification, it was also shown that no sub-harmonic distortion is present in the inverter output. The fundamental period resolution isdependent on the clock-frequency of the micro-controller, and is independent of the numberof samples per period.The second discrete signal processing contribution presented in this thesis was the appli-cation of moving average filter to the single-phase PFC voltage feedback loop. The MAF wasanalyzed and compared to existing digital filtering techniques for single-phase power factorcorrectors. The simple implementation of MAF, along-with computational simplicity, makesMAF a great filter for practical PFC design in low-margin industries. Through experiments535.2. Future Workand simulation, it was shown that through the use of MAF, PFCs respond to load-changes inapproximately half the time that traditional PI controllers do, while maintaining or reducingTHD.The work presented in this thesis was published at IEEE APEC 2013 [2] and IEEE ECCE2013 [3], and submitted for publication [1].5.2 Future WorkThe original contributions in this thesis in the field of multi-carrier sinusoidal synthesis onlyconsider the case of two distinct carrier frequencies. By expanding this technique to includean arbitrary number of distinct carrier frequencies, a new method of generating periodicsignals can be devised that require a minimum number of stored elements. By includingan infinite number of carrier frequencies, with an infinitesimally small resolution, this canalso lead to a method which better analyzes variable-frequency controllers such as hysteresisor bang-bang PWM controllers. 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