Introduction_to_Pulse_Width_Modulated_Control_Systems_1_

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Introduction to PWM control systems

The next few pages describe a novel approach to the study of Pulse Width Modulated (PWM)control systems and provide a systematic way to a general solution of such and similar problems.

Moreover it provides a designer with the ability to predict the system behavior. Consider asimple system consisting of a pulse width modulator (PWM) and a linear system section W(s)(where s is the complex frequency to connect time and frequency domain via the Laplacetransform. [1]; [2]) connected in series. After the individual components are studiedindependently, we will proceed to analyze the system when the feedback loop is closed.

System with symmetrical Pulse Width Modulator

Pulse Width Modulator (PWM)

There is a wide variety of PW modulations; some are shown in Figure 1. In Figure 1, graph A

represents a PWM input time function that is sampled at equal time intervals T. The samplingtime period T, selected by the system designer, is the first parameter defining PWM.

The input time function sampled values are h(0), h(T), h(2T) , etc. The duration of the generatedpulse is κ · h(nT). Kappa ( κ ) is a proportionality factor and is a physical PWM feature definedby the designer; it can be a linear value or a function. The pulse duration for this discussion is



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limited to the selected sampling period (T). Once this width of the generated pulse is reached, thePWM is saturated. The generated pulses have unit amplitude, and are then multiplied by anamplification factor A (also a designer-selectable; physical feature of the modulator.) Thesampling period T, factor kappa ( κ ) and amplification factor A fully define the PWM; all of thesefactors are usually designer-defined.

The Leading Form of the PWM impulse series is shown in Figure 1.B. If the generated pulses aremoved to the end of the sampling period, they result in the Following Form of PWM (Figure1.C). If the input time function sampled value is an integral of the input time function in thepreceding period, this defines the Leading Integrated PWM Form (Figure 1.D). If the LeadingForm of the PWM impulse series is delayed for a desired time ΔT, again designer -selectable, theLeading Delayed Form of PWM is obtained. These observations provide an idea of how manydifferent forms of PWM can be developed – limited only by the requirements of the controlapplication and the designer’s imagination. The variety is practically limitless.

This article considers only the Leading Form of PWM, defined by the sampling period T, factor

kappa ( κ ) and the amplification factor A.

Linear system section W (s)

The linear system section W(s) should not exhibit jumps on the output side when impulses areapplied on the input side. (This would be the case if, for example, a capacitor were connectedbetween the input and output of the linear system section). It is desirable to be able to describethe behavior of the linear system section mathematically if computerized instead of pragmaticanalysis is desired.

Assume a continuous PWM impulse series with impulses of amplitude A, constant width(κ·|x(φ)|) and having interchangeably positive and negative pulses as shown in Figure 2 isapplied to the input of the linear system section W(s). After a reasonable time, the output



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function of the linear system section will stabilize. If the applied function has the period T andis sampled at a rate of T/2, at the sampling time the output function will stabilize to some value|z(φ)| , shown in Figure 3.

This sampled value will influence the generation of the next PWM impulse when the feedback loop is closed. In Figure 3, the applied impulse width is shown in normalized form (divided byT/2). The applied impulse width is represented on the vertical axis and the value of the linearsystem output function at the instant of sampling is represented on the horizontal axis. Becausethe largest pulse width of the applied impulse series to the input of the linear system section isT/2, the largest value of “a” in normalized form is:

( ) / 21

/ 2 / 2

x T a

T T

If the impulse width of the applied impulse series is varied from 0 to T/2, the result is a graph

that starts at the origin and ends on the horizontal line a=1 representing condition when theimpulse series reaches a width of T/2. This curve is a physical feature of the linear systemsection that depends on the sampling time T/2 and selected impulse amplitude A. This is one of the “System Characteristics” of the linear system section.

If the sampling time T/2 is changed and the impulse amplitude A stays the same, another SystemCharacteristic representing the linear system section for the new sampling time is obtained. Byvarying the sampling time of the applied system impulse series, a family of SystemCharacteristics curves representing the linear system section are obtained. These curves definethe behavior of the linear system section when a series of described impulses having differentsampling times and the same amplitude A, are applied to it.

The relationship between the simple system oscillation and the system output function representthe essence of this curve. To provide an example we will examine a linear system section that



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has a frequency transfer function in s plane expressed by:k

W(s) =(T s + 1) (T s + 1)1 2

Initially, assume we select values for this linear system section are T 5.01 seconds,

T 0.52 seconds and the amplification factor k = 1.0. For this case we choose that the PWM willproduce impulses with amplitude A = 2.00. The computed results are shown in Cartesiancoordinate system in Figure 4. On the horizontal axis is the sampled value of the system outputtime function –z(π/ω) . The applied impulses ’ width is plotted in normalized form (divided byT/2) on the vertical axis. For the oscillating case if the period is T, the sampling time is T/2. Thevalue a (normalized impulses width) is:

2 ( )xa

T

The value κ·|x(φ)| represents the width of generated impulses if the sampled value of the inputfunction before the PWM is ( )x . Because the ordinate values are normalized, we can plot

linear system characteristics for different sampling frequencies in same diagram. The systemcharacteristics of selected linear system for sampling times between 0.5 and 5.0 seconds (T=1.0sec to 10.0 sec) are presented in Figure 4. The developed family of System Characteristics canthen be used for analysis of the PWM system when the feedback loop is closed.



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PWM in a control system with closed feedback loop.

The PWM Characteristics and the System Characteristics can be drawn on the same graph, asshown in Figure 5. The dark blue curve is the linear system characteristic from Figure 4 whenthe PWM samples every 4.5 seconds (T=9.0 s). The applied impulse series has amplitude 2.00.

The red line represent the PWM characteristics. The PWM characteristics is symmetrical i.e.positive and negative sampled values are evaluated in same way.

Let’s examine the system characteristic in combination with the PWM (below the linear systemcharacteristic in the Figure 5). System testing is done by exciting the linear system section in astable state with 100 pulses, after which outside excitation is removed and the feedback loop isclosed. The system behavior is observed for 20 additional sampling times. This procedure wasrepeated for 9 different values of excitation varying from 0.05·(T/2) to 0.85·(T/2). Once thefeedback loop is closed and the outside influence is removed, the system stabilizes at the zeroposition. This system is fully stable: experimental results are graphed in Figure 6, which shows

the system output function for three different excitations: smallest (black), medium (red) andlargest (blue).



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If the same system is observed with another PWM, shown in Figure 7 (red line, above thesystem characteristics) the system turns out to be totally unstable; output function is shown inFigure 8, even though the observed PWM is very similar to the PWM considered before.

Now let’s observe the system shown in Figure 9; the linear system characteristic is same asbefore (dark blue line). The pulse width modulator is in the form of steps with a slightly slanted



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horizontal section (red line). At every step it intersects the linear system characteristics goingfrom one to the other side of it and reverse. This system was tested in same way as describedbefore; after system was excited the excitation was removed and the feedback loop was closed.The system performs multiple stable oscillations at the points where the PWM characteristiccrosses the linear system characteristics from upper to lower side. The horizontal part of PWMcharacteristics is deliberately slanted for better observation. The results are shown in Figures 10.



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System with Unsymmetrical Pulse Width Modulator .

Let’s consider a system consisting of an uns ymmetrical PWM in series with a linear systemsection. The modulator could have a curved or slanted line on the positive side that is verydifferent from the one on the negative side, meaning that the modulator itself would evaluatepositive samples differently than negative samples. The positive sample-generated impulselength might be longer or shorter than for the same negative sampled values.

Linear system characteristics enabling system analysis with unsymmetrical PWM.

To be able to analyze such a system and predict its behavior, the linear system section W(s) hasto be prepared for this analysis.

The basic system characteristic is obtained by applying a symmetrical impulse series to the linearsystem section W (s), as described before. For this study the same linear section systemcharacteristic is used; curve T=9.0 sec in Figure 4.

If at one point on the System Characteristics the width of the positive pulses is increased by 5%and the negative pulse width is decreased by 5%, after the system is stabilized two differentvalues are obtained when the linear system output function is sampled. The single point on thesystem characteristic has split into two points. If this is performed for each point on the originalsystem characteristics, two new characteristics curves designated as 0.1·X are generated on eitherside of the original system characteristics. By increasing this asymmetry by 10% another twocurves are obtained, designated as 0.2·X. The slanted lines (brown) in Figure 10 indicatepositions of newly created points from the basic point of original system characteristic. These



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lines are termed the second oscillation boundary of the PWM control system. The graphs inFigure 11 are obtained by increasing the asymmetry of the applied PWM impulse series on thelinear system section.

Unsymmetrical PWM in a control system with a closed feedback loop.

Consider the system curves obtained when the applied impulse series was changed for 10%,shown in Figure 11. These curves are designated as 0.2·X and are shown as separate graphs in



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Figure 12. Let’s combine this system with unsymmetrical PWM, dr awn in the same graph in redcolor. Both sides of the PWM are below the system characteristics curve 0.2·X. Theunsymmetrical PWM used in this case is shown separately in Figure 13. Exciting the system atindividual excitation levels as described before, then closing the feedback loop and removing theinput excitation, the system proves to be stable in all regions as shown in Figure 14.

Similarly, in Figure 15 are the same linear system characteristics used before, combined withunsymmetrical PWM shown with red lines. In this case both sides of the PWM lines are aboveunsymmetrical system characteristics. The PWM characteristics used in this case is shownseparately in Figure 16. This system is treated during testing as described before; it proves to betotally unstable, shown in Figure 17.



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At this time let’s try a more complex system where the characteristic of the PWM intersects theunsymmetrical system characteristics of the linear system part. The unsymmetrical systemcharacteristics used are the same as before, however the PWM is the semi-step function shown insame diagram drown in red, Figure 18. Here again, the horizontal part of PWM characteristics isdeliberately slanted, for more accurate observation. The designed PWM characteristic is shownseparately in Figure 19.



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Exciting the system at individual excitation levels as described before, then closing the feedback loop and removing the input excitation, produces stable system oscillation at different steps of PWM characteristics, presented in Figure 20. It is noticeable that all of these oscillations have apositive DC component. In the case where the unsymmetrical PWM changes sides, the systemwill produce the same oscillation as shown, mirrored across the horizontal axis: oscillations willbe pulled to negative side and will have a negative DC component.



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The last example, shown in Figures 21, has symmetrical PWM with a negative slope in themiddle. The PWM characteristic is shown separately in Figure 22. This PWM characteristicsnegative slope is steeper than the system brown lines representing the se lected system’s secondoscillation boundaries.



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The system is tested as before, and after the excitation is removed the system settles into one of the two stable oscillation states, Figure 23 and Figure 24. One of these oscillations exhibits apositive DC component and the other a negative DC component. What is remarkable in thissystem is that it produces two stable oscillating states, one with a positive and one with negativeDC components. Conceivably, with some outside influence it could move from one oscillatingcondition to the other, acting in some ways like a storage element.



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If the PWM characteristics negative slope was chosen to have a slant smaller than the brownlines in diagram 21, once the feedback loop is closed, the system would stabilize and performsimple stable oscillation in the point where the PWM characteristic crosses the basic systemcharacteristics (T=9.0 sec.).

We have chosen a simple system to illustrate a fundamental approach to study of such systems.Different systems exhibit different behaviors from very complex to those presented here. In anycase it is up to the design engineer to choose and adjust system parameters to meet his or hersrequirements. The examples given herein suggest the possibility of extremely precise systemcontrol which, if applied for example to control of missiles, could render presently knowndefense systems obsolete.



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This concludes the outline of a basic foundation for analysis and solution of PWM (and similar)control systems. Readers are encouraged to extend and apply the foregoing methodology toaddress the demands of their specific PWM applications.

Zelimir Krokar, Boston

Copyright June, 2009 by Zelimir Krokar.

Reproduction or publication of the content in any manner, without express permission of theauthor, is prohibited. No liability is assumed or accepted with respect to the use of theinformation herein.

Refferences:

1. Gustav Doetcsh: Anleitung zur praktschen Gebrauch der Laplace-Transformation; R.Oldenbourg, München 1961

2. John G. Truxal: Automatic Feedback Control System Synthesis; McGraw-Hill Book Company Inc., New York, Toronto, London 1955

3. Zelimir Krokar: Theory of Pulse Width Modulated Control Systems (not published,available on request)

4. Angel V. Peterchev: Digital Pulse-Width Modulation Control in Power ElectronicCircuits: Theory and Application, Berkeley 2005

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