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The Mathematical Foundation of Distributed Interleaved Systems Shaul Ozeri, Doron Shmilovitz, Member, IEEE, Sigmond Singer, Member, IEEE, and Luis Martinez-Salamero
Abstract—The distribution and interleaving (D&I) of signals is a common method for ripple attenuation in various engineering applications in such areas as control, communication, and power electronics. Similarities to this technique may also been found in nonengineering fields such as biology and medicine. This paper presents a mathematical exploration of distributed interleaved systems along with a simple frequency-domain model of interleaving. We are hoping that the insights provided by this mathematical framework and the newly proposed model for interleaved systems will lead to enhanced techniques for evaluating D&I processes, and facilitate the design of better systems. In particular, we hope this work results in new approaches to low-pass filtering that will exhibit fast dynamics and very efficient ripple attenuation (in theory, this can produce complete ripple removal in some cases). Index Terms—Averaging, distributed system, dynamics, electromagnetic interference (EMI), interleaving, pulsewidth modulation (PWM), ripple, switched-mode power converter.
I. INTRODUCTION
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HERE are many situations in technology where the average value of pulsating waveforms is needed. In some instances, the pulses produced by a source are applied to a system that requires that their value be averaged; on other occasions, the source produces constant value amplitude, but the inherent functioning of the absorbing system generates pulses. Pulse-generating sources are very common in communication and control systems. Interestingly, a similar issue exists in medicine. In many cases, administrating a drug at a constant (basal) rate is preferable to administering the drug in pulses via a bolus [1], [2]. Switched mode converters, Class-D amplifiers, laser systems, and control systems are other examples of systems that generate pulses in the course of their normal functioning. The application of low-pass filters to extract the average value of pulsating waveforms is quite common, as it removes the ac component to some extent. This method of averaging is attractive due to its simplicity, but the attendant disadvantages are the large amount of storage capacity needed to obtain good filtering, and the resulting slow response to control command. This arises from the tradeoff between the quality of the filtering Manuscript received August 3, 2005; revised February 28, 2006. This work was supported in part by the Israeli Science Foundation (ISF) under Grant 1250/05. This paper was recommended by Associate Editor M. K. Kazimierczuk. S. Ozeri, D. Shmilovitz, and S. Singer are with the School of Electrical Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel. (e-mail:
[email protected]. ac.il;
[email protected];
[email protected]). L. Martinez-Salamero is with Departamento de Ingeniería Eléctrónica, Escuela Técnica Superior de Ingeniería, University Rovira i Virgili, Tarragona 43007, Spain (e-mail:
[email protected]). Digital Object Identifier 10.1109/TCSI.2006.886001
and the dynamics–the lower the cutoff frequency, the slower the system’s response. To correct this problem, an alternative method for averaging pulses has been developed. A train of narrow pulses is averaged by applying them to a network of transmission line segments of different lengths prior to applying them to the load where their energy is aggregated. Because of the differing lengths of the transmission-line segments, averaging is achieved by spreading the pulses in time [3]. Current pulses generated by switched-mode converters can be averaged by replacing a single converter with a group of identical converters, each of which would process of the total power. These converters are operated synchronously with a constant time shift from one to the other. This method is known as the interleaving of operating converters [4]–[9]. To relate this to the framework of medicine, single large injections are often replaced by multiple small-volume injections spread out over time, as this results in a smother concentration profile in the blood [1], [2]. Other solutions include sustained drug release mechanisms. The examples mentioned above demonstrate that averaging can be achieved by reconstructing the total system so that it consists of a network made up of elements that process the pulsating signals. The distribution of pulses among processing elements and the synchronized time shift in their operation can be achieved either by means of control (such as power converters operating in an interleaved mode [4]–[9]) or through the natural properties of the system (as in the case of transmission line-based averaging [3]). As the desired averaging is achieved by arranging the functional elements in a suitable topology, the operation is viewed as distributed processing. Usually, the modification of the original system involves splitting up of the pulsating parts among small units. For example, in a system consisting of a source, a pulsewidth modulated (PWM) converter and a load, the single converter is actually divided into smaller converter units (actually replaced by smaller converters). Even though the converters operate synchronously, it is advantageous to shift their operation in time one from the other, resulting in a distributed and interleaved (D&I) system. The application of interleaving techniques to a 1.5-kW power factor, corrected rectifier was presented in [4], and demonstrated advantages in terms of efficiency and electromagnetic interference (EMI). Cellular power-conversion architecture was presented in [5], and indicated potential advantages in terms of reliability and cost, and offered new solutions to challenges in practical design of such systems. Paralleled interleaved converters were explored in [6], which indicated the existence of
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upper limits to ripple attenuation. The stability of paralleled converters was investigated in [7] where small-signal modeling was employed. A general methodology for distributed interleaving was introduced in [8], which indicated the possibility of implementation of a varying number of converter cells without centralized control and automatic accommodation. A performance comparison under different clocking regimes was also provided in [8]. Further investigation of interleaved power conversion and its application to Forward converters was presented in [9]. It has been found that the quality of the averaging (ripple reduction), as well as the required amount of storage, is a function of the number of elements ( ) that the D&I network comprises. The larger the , the better the averaging and the smaller the storage required. As tends to infinity, ripple and required storage approaches zero. Due to the reduced storage requirements, a system that includes a D&I network has much faster dynamics compared to one that employs a conventional low-pass filter to achieve averaging. This paper provides a mathematical treatment of D&I through alternate representations in time and frequency domains. The suggested treatment offers insights into the interleaving action as well as design considerations for D&I systems. In particular, due to the resince there is a practical limit to the size of sulting increased complexity, a design approach is proposed for determining the optimal value of . Switched mode converters are often used as examples of D&I applications for the purpose of averaging and as a method to reduce ripple and storage requirements. However, it needs to be stressed that the results of this research are general and have application to a broader range of circuits and systems.
Equation (4) represents the transformation that a signal undergoes due to D&I action, resulting in the D&I output signal . Applying Fourier transform yields
(5)
The D&I action is thus linear, and in the frequency domain can be represented as a multiplying operator
(6) is periodic with a period , it can Since the initial signal be expanded by a Fourier series, i.e., it has nonzero values only . It is therefore at the harmonics of the fundamental, only at the harmonic frequencies meaningful to evaluate defined in (7) where
is the harmonic number. Substituting (7) in (6) yields (8)
II. FREQUENCY-DOMAIN FORMULATION Let us begin by considering a periodic signal by an interleaved system
processed
(1) (2) is its Fourier transform. where is the signal period and is the angular frequency . Assume that is processed by an interleaved multiple system consisting of identical elements. will be referred to as the degree of distribution. The signal is first divided by so as to maintain the overall signal power. Due to symmetry considerations, it is assumed that each element implies identical time shifts, , from one to the next. Thus, each system element generates a signal of the form
acSubstituting the appropriate delay defined in (3), quires the form described in (9). The appropriate delay is aswhere is the delay introduced sumed to be by the th channel, the basic signal’s period and the degree of distribution. Though it can be mathematically proved that the optimal delay in terms of ripple rejection is the one defined in is as(3), this proof is not provided herein and sumed due to symmetry considerations
(9) acquires nonzero values only It can easily be seen that for values of that are integer multiplies of the distribution deequals gree . For these values of , it can be shown that 1 (using L’Hopital’s derivative rule)
(3) The distributed element outputs are then summed up, yielding (4)
,
(10)
This result is summarized in the matrix form of (11) where may be viewed as a weighting function acting on the
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vector of harmonic components of , as shown in (11) at the bottom of the page. is a diagonal sparse matrix of infinite rank, whose elements are defined as , (12) as an infinite vector of harmonics, the Thus, viewing D&I transforms it into a meager series leaving only harmonics of the order of integer multiples of the degree of distribution .
.. . .. .
(13)
Fig. 1. Time-domain representation of D&I systems.
.. . .. .
delay. Finally, the delayed signals are summed up yielding . Fig. 1 resembles finite-impulse response (FIR) digital filters, consisting of successive shift unit delay and summation as in moving average (MA) digital filters with a rectangular window [10], [11], as depicted in Fig. 2. The difference is that FIR filters contain the option of in Fig. 2, (which actuchoosing the weighting coefficients, ally constitutes the FIR filter design), whereas the D&I system represents the distribution of one big power converter, or any other unit that performs a physical work such as a medical pump, into many small converters (or other work generating
III. MODELING AND PHYSICAL INTERPRETATION OF D&I Let us define a delay operator by (14) In doing this, the D&I operation defined by (3) lends itself to the model illustrated in Fig. 1. Accordingly, the signal is first signals with amplitudes reduced by the factor divided into . Next, each of the signals goes through an appropriate
..
.
..
.. . .. . .. . .. .
.
(11)
.. .. . .. . .. . .. .
. .. . .. . .. . .. .
.. . .. . .. . .. .
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Fig. 2. Direct form structure of FIR Filter.
(depending on the spectral content of ) due to the application of the D&I (easing further filtration of the ripple, if needed). (c) Since many frequency components (in particular, the first components) are blocked by D&I, the ripple is attenuated and its energy is significantly decreased. The precise extent to which the energy is reduced depends on the signal’s harmonic composition. IV. EFFECT OF THE SIGNAL’S SHAPE IN D&I SYSTEMS AND SOME DESIGN CONSIDERATIONS FOR D&I SYSTEMS
Fig. 3. Interpretation of D&I action in the frequency domain. (a) Representation as a matched periodic bandpass filter. (b) Representation as a harmonics selective filter.
units). The D&I systems discussed herein actually constitute a reorganization of a power/work processing initial unit (not a dedicated filter unit). Ripple filtering is achieved through this reorganization of the previously existing system, in addition to the initial functioning of the overall system. Thus, the need for dedicated filters in the system is reduced and in some cases completely removed. Thus, as opposed to processing of information signals, we deal with physical signals, and rather than inserting a filter (a physical one), we rearrange the system in a way that yields a filter like behavior. Other related realization are switched-capacitor filters [12], and some concepts in neural networks theory. Modeling the D&I system of Fig. 1 in the frequency domain provides additional insight. Equations (11) and (12) imply that at D&I action does not affect frequency components of , while the rest of frequencies that are integer multiples of are completely removed by this process. the harmonics of This may be represented by a periodic bandpass filter that has (and has no phase no attenuation at frequencies around insertion at these frequencies, either), but has infinite attenuation at all other frequencies. This model is depicted schematically in Fig. 3. It should be stressed that the pass frequencies of this special filter are not constant, but rather depend on the signal’s fundamental frequency, thus implying matched filter modeling. Another way to describe this action is as a harmonics bandpass filter, as depicted in Fig. 3(b). Some inherent properties of the D&I action should be pointed out. (a) The dc component of the signal is not affected by D&I. (b) The lowest frequency component that can possibly pass times the fundamental (prothrough the D&I is . Thus, vided that this frequency component exists in to at least the ripple frequency is increased from
As observed previously, D&I systems act as bandpass filters with very unique features. Their performance depends greatly on the input signal’s frequency composition. For this reason, a D&I system needs to be designed to perform its designated function while, at the same time, taking into consideration the nature of the signal that it will be processing. These issues are elaborated upon in this section. A major purpose of D&I is extraction of the signal’s average and rejection of its ripple. The ripple reduction will thus serve as a benchmark of performance. Since the Fourier coefficients of normally observed waveforms are a decreasing series, filtering by a D&I system may be viewed as a means of accelerating the decrease of harmonics. , is defined in (15) as the sum of its A signal’s ripple, ac components (15) The application of D&I to some particular types of signals is discussed below. In particular, this discussion will include aspects related to design considerations. A. Signals of Finite Bandwidth The ripple of signals that contain a finite number of harmonics can be completely canceled out by D&I. This can be done simply by choosing to be greater than the highest harmonic index contained in the signal. In this case, the first harmonic that could potentially pass through the D&I is higher than any harmonic contained in the signal. For example, if the ripple is of a pure sinusoidal type, D&I of degree 2 is sufficient to reject it completely. This is because D&I blocks all the harmonics up to the second harmonic, while the signal contains only the first harmonic. Of course, if is greater than the minimal value yielded by the highest harmonic contained in the signal, the ripple is still canceled. The return current of a balanced three-phase system may be viewed as an example of a signal of finite bandwidth. The initial signal contains only a first harmonic while the degree of distribution is 3. Therefore, the return current is theoretically zero.
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B. Signals With Half-Wave Odd Symmetry A signal, is said to exhibit half-wave odd symmetry if it obeys (16), and the Fourier series of such signals is known to contain only odd harmonics (16) Fig. 4. Example of a sawtooth waveform.
If the original signal, , to be processed by the D&I exhibits half-wave odd symmetry about its average value [that is, was subtracted from the signal, then if the average value of the signal would exhibit half-wave odd symmetry according to (16)], then the Fourier series of the signal’s ripple consists only of odd harmonics
D. Effect of D&I On Some of the Commonly Observed Waveforms 1) Sawtooth Waveform: Consider the sawtooth signal defined in (20) and illustrated in Fig. 4 for
(17)
Ripple reduction of such signals (signals that contain no even frequency components) by means of D&I is quite simple and efficient. Since such signals contain only odd harmonics, the ripple is completely canceled out provided that is chosen to be even. This is because the D&I only allows frequency com( ) to pass through; ponents of even multiples of in this case, the signal contains no such frequencies (only fre). This is an quencies that are odd multiples of , interesting result since, contrary to intuition, D&I of an even deor produces better ripple attenuation gree such as (theoretically perfect) than a higher that has an odd value. C. Signals That Can Be Represented By Even Harmonics Only Consider a signal whose Fourier series representation contains only even harmonics (in addition to the fundamental) as described as
(20) Representing this signal by its Fourier series shows that this signal has an infinite Fourier series representation that contains both odd and even harmonics. Thus, it does not belong to any of the three special categories mentioned above. Yet, some specific evaluation can be performed. By applying the ripple definition can be ex(15), the ripple of the sawtooth signal pressed in terms of its harmonic components (21)
Where the sign “ ” represents proportionality [and in (21) through (40) it might be used interchangeably for the “ ” sign]. The coefficients (harmonic amplitudes) are seen to decrease proportionally to the harmonic number to the power of ( ). After application of D&I to the sawtooth signal, the dc component, , shows no change, but the ripple changes according to (12). is substituted for . This ripple is expressed in (22), where (22)
(18)
In this case, should be chosen to be an odd number so that the lowest frequency component that passes through the D&I system is calculated as (19) For relatively large values of , the ripple energy is dramatically attenuated since harmonic content is diluted for all fre, (especially quencies in the range , which consince the lowest frequency components up to tribute most of the signal’s energy, are removed). The ratio of ripple attenuation also depends on the signal’s shape. The faster the decrease of the harmonic’s amplitude, , the smaller the required to obtain a given ripple attenuation.
In inspecting (22), it is noted that it is of the same form as (21). Comparing these last two equations, we can conclude the following. a) The ripple after D&I is of the same nature as the ripple at the D&I input, i.e., it is also a sawtooth function. b) The ripple frequency after D&I has increased by a factor with respect to that at the D&I input (facilitating of easier further filtration, if needed). c) The ripple amplitude has decreased by a factor of therefore, its energy has been reduced by . The ripple waveform resulting from a sawtooth waveform after D&I, is displayed in Fig. 5 for 3 values of , where is assumed, without loss of generality. Since the sawtooth signal contains all frequency harmonics and since the harmonic amplitudes decrease linearly with the harmonic index, choosing the degree of distribution is based
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Fig. 5. Ripple produced by a sawtooth waveform treated by D&I systems of degrees 7, 8, and 9.
Fig. 7. Ripple waveforms resulting from a jsin(t)j signal for D&I systems of degrees 7, 8, and 9.
Fig. 6. Normalized, absolute value of a sine wave direct form structure of FIR Filter.
solely on satisfying the ripple reduction requirement according ripple reduction ratio. to the It should be noted that the lower limit of ripple attenuation . For most that can be performed by D&I is proportional to other signal waveforms, the ripple can be reduced even more effectively. This is achieved by analyzing the Fourier series repand selecting a proper value of . resentation of 2) Absolute Value of Sine Waveform: Another example of a commonly observed waveform function which does not belong to any of the above categories is the absolute value of a sine wave, as depicted in Fig. 6 (in which, without loss of generality, the angular frequency and the amplitude are chosen equal to 1).
Fig. 8. Example of triangular waveform.
frequency is doubled compared to the original nonrectified sine wave). Thus, as in the sawtooth case, there is no advantage in choosing an odd or even degree of distribution, . The rate at which the frequency components decrease is approximated as
(23) (25) This kind of waveform is commonly observed in power systems in which it is produced by full wave rectification of sinusoidal voltages. The Fourier series expansion of this signal is given by
(24) This waveform might incorrectly be assumed to be one of those with only even harmonic expansion. However, in comparing (24) to (18), it is seen that this expansion actually contains both odd and even harmonics (except that the fundamental
This high rate decrease implies that efficient ripple cancellation can be achieved even with relatively low values of . Fig. 7 presents the ripple waveform resulting from a signal after D&I for values of 7, 8, and 9. A relatively big difference is noted between the tracings. (compare this to Fig. 5 where the difference is smaller). Also, the ripple is reduced at a much faster rate as compared to the attenuation for a sawtooth waveform, as shown in Fig. 5. In the following waveform examples, it is seen that a more yields much more efficient ripple propitious choice of attenuation. 3) Triangular Waveform: Consider the triangular waveform displayed in Fig. 8. By inspection, it can be observed that this is a signal with half-wave odd symmetry.
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Fig. 9. Total ripple cancellation for a triangular waveform due to half-wave odd symmetry.
Indeed, the triangular waveform contains only odd harmonics, as it can be seen from (26). This signal’s ripple is proportional to the series expansion of
(26)
in this case would be an even value. The best choice for This would result in a D&I action that would theoretically cancel the ripple completely and extract the signal’s average. The sim, as illustrated in Fig. 9. Obviously, plest choice would be the ripple is completely removed in this case. If is odd, the harmonics that pass through D&I are at mul( in this example). Thus, the D&I produces tiples of a signal similar in form to the one described in (22)
(27)
Rearranging (27), and recalling that yields
is an odd number,
(28)
In inspecting (28), one realizes that it has the same form as (26), implying the following. a) The ripple after D&I remains triangular. b) The ripple frequency is times higher. ; c) The ripple amplitude is reduced by a factor of . therefore, its energy has been reduced by The ripple attenuation ratio is depicted in Fig. 10 for different values of the interleaving degree . Therefore, when designing a D&I system for an odd function, should be chosen to be even. Even though with the ripple is completely canceled theoretically, it may be advantageous to use higher even values of due to practical constraints such as sensitivity to parameter variation (an important issue which is beyond the scope of this article), distribution of
heat sources, component rating in the case of power conversion, and so on. If is not chosen to be even, the D&I performance is similar to the performance achievable in the case of an absolute-sine type signal and much better than the performance in the case of a sawtooth signal. 4) Chopped Sine Waveform: Another example of a half-wave odd symmetry signal is the chopped sine waveform depicted in Fig. 11. Such waveforms are observed in triac-based dimmer circuits that regulate the power flow to resistive loads such as incandescent light bulbs (providing an efficient way to control the load power by allowing only a portion of the 50/60 Hz current to pass through). This is evidently a half-wave odd symmetry waveform whose ripple may be totally removed by applying D&I of an even degree, as depicted schematically in Fig. 11(b). V. ZERO-RIPPLE OPERATION WITH PWM RECTANGULAR WAVEFORM A PWM rectangular waveform is a very common type of signal in engineering applications [13]–[21]. One area in which PWM modulation is employed in data communication is in the digital subscriber loop (DSL) modems. A typical analog front-end of a DSL modem is illustrated in Fig. 12. The transmit and receive channels whose operations are separated in frequency are connected in parallel and coupled to the telephone line by a line coupling transformer. The received signal level depends on the transmission line length, which can vary from 11 to a few hundred meters. Consequently, the received signal level can vary greatly and may be as low as a few millivolts. In order to achieve full digitizing resolution, the signal is usually amplified by an amplifier with automatic gain control (AGC). The amplifier’s gain is controlled by the modem’s controller, in Fig. 12). usually by an analog signal ( Since transmission line characteristics do not change very , varies at a slow quickly, the AGC programming signal, rate. In many cases, a PWM signal (generated by the controller) . This is used to generate the analog programming signal, is actually a kind of serial, low rate analog data communication, but it satisfies the required rate and saves on the number of pins needed. Prior to being applied to the AGC amplifier, the PWM signal is low-pass filtered in order to generate the programming . This low-pass filter is a critical element of the voltage, modem. On one hand, filtration requirements (ripple rejection) are very severe in order to prevent injection of noise into the receive channel. On the other hand, this filter must not introduce are important in order too much delay; the dynamics of to quickly respond to line variations (such as impedance and attenuation variations caused by other subscribers using the line). If the dynamics are not good enough, data will be lost during these transitions. So, quite sophisticated (and complex) filters are required. The suggested D&I method can replace this filter achieving (as will be shown in this section) total ripple attenuation (in principle) and inserting a delay on the order of the PWM signal’s period (which is typically in the range of a few microseconds, 2-3 orders of magnitude smaller than the delay inserted by a conventionally employed low-pass filter). In the field of power electronics, this mechanism is the most widely employed modulation method for controlling switched-
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Fig. 10. Ripple attenuation ratio for triangular waveforms.
The amplitude of these coefficients is therefore bounded according to (32) It is therefore concluded that, the higher the , the better the harmonics will be ripple rejection (as the first blocked). It will be shown that in many practical systems they is defined can be completely canceled out. The duty cycle as the percentage of time that the signal is high, or its numeric value may also be used (33) Fig. 11. (a) Chopped sine waveform and (b) ripple removal with D&I of degree = 2.
N
mode converters [13]–[18] and motor drives [19], and in communication and control systems, it is used for data modulation [20], [21]. This type of waveform is used in a wide range of applications, and therefore, deserves special attention. This signal is defined in (29) and its Fourier series representation is given by the following (see Fig. 13): for for
and (29) (30)
Although there are several methods used to modulate information in a PWM signal, the most commonly used one is to keep the frequency constant and modulate the “on time” component, effectively resulting in the modulation of the duty ratio. This modulation method is assumed for the following discussion, but this analysis applies regardless of the particular type of PWM used. From (29) it can be seen that a PWM signal with a 50% duty cycle exhibits half-wave odd symmetry. Indeed, substitution of in (33) results in the removal of all of the even harmonic components. Choosing an even degree of distribution would result in total ripple cancellation. This is, however, just an isolated value within the duty cycle. In practical engineering systems, the information is modulated over the duty ratio, implying a time-varying quantity duty ratio (34)
It can be seen from (30) that the Fourier series coefficients behave as
(31)
Let us suppose it is possible to find suitable values for that will eliminate the ripple. These values of will, of course, depend on the value of zero ripple
(35)
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Fig. 12. Typical analog front-end of a DSL modem.
By choosing the D&I degree, , to be equal to the number of discrete levels of (or ), yields (39)
Fig. 13. PWM rectangular waveform.
Since varies, it would be required to find a distribution degree that solves (33) for all values of (another option would involve a D&I system with an adaptive distribution degree, but this option is not explored in this paper). According to (30), rectangular waveforms contain all of the harmonics to infinity and, in the general case, have both odd and even harmonics. This implies that (35) cannot be satisfied for continuous values of . Nevertheless, it can be shown that in most practical engineering systems involving PWM signals, (35) can be satisfied, provided that assumes only discrete values. Digital control is becoming very widely employed nowadays in nearly all areas in which control is applied, and the same is true with respect to digital communication systems. In such can acquire only discrete values, and the duty ratio systems, may therefore be represented as a discrete quantity
(36) where is the number of discrete levels (practically representing the quantization resolution of ). It should be stressed that must be a rational number (for instance, if the hardware can generate duty cycle values with a resolution of up to 1%, ). then Since the duty ratio has discrete values, so does (37) Substituting the discrete value of
in (31) yields
(38)
According to (10), applying D&I to a signal having the coefficient of (39) blocks all the frequency components except those that are equal to an integer multiple of the fundamental, . This implies that the only frequency components left after D&I have the coefficients defined in (40), which are seen to be equal to zero for all (40) It is therefore concluded that the ripple of a PWM signal with a quantized duty ratio can be completely removed by D&I, provided that the degree of D&I is an integer multiple of the number of quantized values. VI. CONCLUSION The effect of D&I has been mathematically explored and discussed within the context of practical engineering applications. Simple modeling in terms of a periodic bandpass filter was suggested which furthers our understanding of the operation of D&I systems. It is concluded that, in general, D&I is an effective tool for averaging periodic wave signals and rejecting their ac components, without involving the drawbacks inherent in conventional low-pass filters such as deterioration in dynamics and the need for bulky storage components. It was demonstrated that the performance of D&I systems depends greatly on the nature of the signal being processed, and that this performance can be dramatically improved by proper design of the D&I system. The performance of D&I systems has been shown to vary significantly from one signal to the other, depending on the waveform’s properties, including its spectral composition. It is hoped that the extensive discussion of design considerations will lead to improvements in the performance of D&I systems in the future. Further research in needed to develop specific criteria for evaluating ripple attenuation in terms of peak-to-peak voltage (or current), overall ripple power, and specific harmonic power, among other parameters.
OZERI et al.: DISTRIBUTED INTERLEAVED SYSTEMS
REFERENCES [1] J. Urquhart, “Controlled drug delivery: Therapeutic and pharmacological aspects,” J. Internal Medicine, vol. 248, pp. 357–376, 2000. [2] S. Lee and E. Hitt, “Continuous subcutaneous insulin infusion: Intensive treatment, flexible lyfistyle,” Clinical Update, Aug. 2003. [3] D. Shmilovitz and S. Singer, “Current averaging networks based on transmission lines,” in Proc. IEEE ISCAS’95, Seattle, WA, May 95, pp. 2181–2184. [4] B. A. Miwa, D. M. Otten, and M. F. Schlecht, “High efficiency power factor correction using interleaving techniques,” in Proc. IEEE Appl. Power Electron. Conf., Boston, MA, 1992, pp. 557–568. [5] J. G. Kassakian and D. J. Perreault, “An assessment of cellular architectures for large converter systems,” in Proc. Int. Power Electron. Motion Contr. Conf., Beijing, China, 1994, pp. 70–79. [6] C. Chang and M. Knights, “Interleaving technique in distributed power-conversion systems,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 42, no. 5, pp. 245–251, May 1995. [7] V. J. Thottuvelil and G. C. Verghese, “Stability analysis of paralleled DC/DC converters with active current sharing,” in Proc. IEEE Power Electron. Special. Conf., 1996, pp. 1080–1086. [8] D. J. Perreault and J. G. Kassakian, “Distributed interleaving of paralleled power converters,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 44, no. 8, pp. 728–734, Aug. 1997. [9] M. T. Zhang, M. M. Jovanovic, and F. C. Y. Lee, “Analysis and evaluation of interleaving techniques in forward converters,” IEEE Trans. Power Electronics, vol. 4, p. 690, 1998. [10] A. V. Oppenheim and A. S. Willsky, Signals and Systems. Englewood Cliffs, NJ: Prentice-Hall, 1983. [11] J. G. Proakis and D. G. Manolakis, Digital Signal Processing, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 1996. [12] R. G. Gregorian and W. E. Nicholson, “Switched-capacitor decimation and interpolation circuits,” IEEE Trans. Circuits Syst., vol. CAS-27, no. 6, pp. 509–514, Jun. 1980. [13] G. W. Wester and R. D. Middelbrook, “Low-frequency characterization of switched dc–dc converters,” IEEE Trans. Circuits Syst., vol. CAS-21, no. 5, pp. 376–385, May 1973. [14] V. Vorperian, “Simplified analysis of PWM converters using the model of the PWM switch: Parts I & II,” IEEE Trans. Aerosp., Electron. Syst., vol. 4, pp. 205–214, Apr. 1989. [15] S. R. Sanders and G. C. Vergese, “Synthesis of averaged circuit models for switched power converters,” IEEE Trans. Circuits Syst., vol. 38, no. 8, pp. 905–915, Aug. 1991. [16] D. Czarkowski and M. K. Kazimierczuk, “Static- and dynamic-circuit models of PWM buck-derived dc–dc convertors,” IEE Proceedings G, Circuits, Devices and Syst., vol. 139, no. 6, pp. 669–679, Dec. 1992. [17] R. W. Erickson and D. Maksimovic, Fundamentals of Power Electronics, 2nd ed. : Kluwer academic Publisher, 2001, pp. 413–420. [18] V. Vorperian, Fast Analytical Techniques for Electrical and Electronic Circuits. Cambridge, U.K.: Cambridge University Press, 2002. [19] N. Mohan, T. M. Undeland, and W. P. Robbins, Power Electronics, Converters, Applications and Design, second ed. New York: Wiley, 1995. [20] “RF system and circuit challenges for WiMAX,” Intel Technol. J., vol. 08, no. 03, Aug. 2004. [21] “ATSC QAM NTSC BTSC FM front-end,” Oren Semiconductor Product Brief, Dec. 2003.
Shaul Ozeri received the B.Sc. and M.Sc. degrees from Tel-Aviv University, Tel-Aviv, Israel, in 1987 and 2002, respectively. He is working toward the Ph.D. degree, working on high-frequency distributed acoustical projectors for medical applications. From 1987 to 2000, he worked in the data-com industry as a Senior Designer of Ethernet 802.3-based products and VDSL modems. Since 2000, he has been working in the biomedical industry, developing advanced drug delivery systems and transdermal drug delivery systems based on a hybrid piezo-magnet technology for the diabetic market.
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D. Shmilovitz (M’98) was born in Romania in 1963. He received the B.Sc., M.Sc., and Ph.D. degrees from Tel-Aviv University, Tel-Aviv, Israel, in 1986, 1993, and 1997, respectively, all in electrical engineering. During 1986-1990, he worked in R&D for the IAF, where he developed programmable electronic loads. During 1997–1999, he was a Post-Doctorate Fellow at New York Polytechnic University, Brooklyn, where he worked on unity power factor bidirectional onboard chargers for electric vehicles. Since 2000, he has been with the faculty of Engineering, Tel-Aviv University, where he established a state-of-the-art power electronics and power quality research laboratory. His research interests include switched-mode converters: topology, dynamics and control, power quality and power conversion for alternative energy sources, and general circuit theory.
Sigmond Singer (M’92) received the B.Sc. and D.Sc. degrees from the Technion, Haifa, Israel, in 1967 and 1973, respectively. In 1978, he joined the staff of the Faculty of Engineering, Tel-Aviv University, Tel-Aviv, Israel, where he is currently a Professor. During 2000 to 2004, he served as the Chairman of the Department of Interdisciplinary Studies at Tel-Aviv University. His research interests are general circuits and systems theory, power electronics, and energy conversion. Dr. Singer was the recipient of the IEEE Circuits and Systems Society’s Best Paper Award Committee 1990 Darlington Award, for his paper “Realization of Loss Free Resistive Elements.” He is currently the Chairman of the IEEE section in Israel.
Luis Martínez-Salamero received the Ingeniero de Telecomunicación and the doctorate degrees from the Universidad Politécnica de Cataluña, Barcelona, Spain in 1978 and 1984, respectively. From 1978 to 1992, he taught circuit theory, analog electronics, and power processing at Escuela Técnica Superior de Ingenieros de Telecomunicación de Barcelona, Barcelona, Spain. During the academic year 1992–1993 he was a Visiting Professor at the Center for Solid-State Power Conditioning and Control, Deparment of Electrical Engineering, Duke University, Durham, NC. He is currently a Full Professor in the Departamento de Ingeniería Eléctrónica, Eléctrica y Automática, Escuela Técnica Superior de Ingeniería, Universidad Rovira i Virgili, Tarragona, Spain. During the academic year 2003–2004, he was Visiting Scholar at the Laboratoire d’ Architecture et d’Analyse des Systèmes ( L.A.A.S), Research National Center ( CNRS), Toulouse, France. His research interest are in the field of structure and control of power conditioning systems for autonomous systems. He has published a great number of papers in scientific journals and conference proceedings and holds a US patent on the electric energy distribution in vehicles by means of a bidirectional dc-to-dc switching converter. He is the Director of the Grupo de Automática y Electrónica Industrial, a research group on Industrial Electronics and Automatic Control whose main research fields are power conditioning for vehicles, satellites and renewable energy. Dr Martínez-Salamero was Guest Editor of the IEEE TRANS. CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS for the August 1997 Special issue on Simulation, Theory and Design of Switched-Analog Networks. He has been Distinguished Lecturer of the IEEE Circuits and Systems Society in the period 2001-2002. He is currently the president of the Spanish Chapter of the IEEE Power Electronics Society
