Rectifier Design and Analysis - BWC Electronics

1 Rectifier Design and AnalysisRectifier Design and AnalysisDisclaimer:This document, associated technical descriptions and Design information comprise a W5 BWC Electronics project done exclusively for John L. KeithW5 BWC. This is an original work of W5 BWC Electronics intended to function properly and be accurately presented as described herein; however, nopart of this project is offered for sale, presented to be free of patent infringements, or represented to be fit for any particular use. Any public use of thisinformation is offered for educational purposes only, as a description of a personal project. Any and all liability of its' use is the sole responsibility of 2009 by W5 BWC ElectronicsWP10A190 Rev BThis document describes the operation and Design of AC line operated rectifiercircuits. Both theory of operation and detailed Design procedures are included,neither of which require complex differential equations for Analysis of the tran-scendental this is not a mathematically "pure" Analysis , it is based on the principles oftranscendental relationships using graphical integration, initial conditions andsolution of discontinuous functions; providing useable results for the actualconstruction of working systems as well as an intuitive understanding of thevoltages and currents involved.

2 The complex mathematical Analysis normallyrequired to predict circuit behavior has left many, otherwise technically savoy,individuals relying on handbook data without understanding the actual interac-tions document is written for individuals with a technical understanding of basicelectronics. = 0t = 0 T =//3/2tototftftc0tc0'tc00-iC iC iOif ve vC Ve vC(to) vC(min) vC(tf) vC(max)CICOtCICOtc0'W5 BWC Electronics9108 FM 1972 Gilmer, TX 75645 February 20092 Rectifier Design and AnalysisTopicPageIntroduction to Rectifier Design FWCT FWBR Model FWCT dual outputRS3 Voltage and current relationships conventions used in document6 Qualitative Analysis6 Handbook Factors8 First Pass Approximations9 Quantitative Analysis12 RMS Calculations15 Component Selection17 AppendixA Derivation of equations20B Related equations and Discussion22C Fourier Series for FW Rectifier24D Example Excel Spreadsheet for Figure 1025E Excel Spreadsheet instructions26F Components31 Contents.

3 W5 BWC Electronics9108 FM 1972 Gilmer, TX 75645 February 20093 Rectifier Design and AnalysisThe objective of this document is to provide an intuitive approach to rectifierdesign that will enable the reader to grasp the complex relationships of voltageand currents in these seemingly simple circuits. Taking this intuitive under-standing to the next step, a practical approach is presented to calculate (withina reasonable degree of accuracy) the peak andrms voltage and currents neces-sary to specify transformers, diodes and capacitors for a given , Electrical Engineering texts present basic Rectifier Analysis usingtranscendental relationships solved with differential ,2 But, theinclusion of real world conditions can make this Analysis worthy of a Doctorialpaper. Presented here is an approach that accurately designs Rectifier systemsproducing both useable results and an understanding of the voltages andcurrents involved while being simple enough to be understood by a person witha basic technical background in Electronics .

4 However, this Analysis is based onthe same transcendental relationships, typical of time variant systems, but theequations presented or either developed from basic electrical laws that thereader can follow or referenced to published technical the reader is assumed to be familiar with basic electronic theory, thisdocument will limit further explanations to rather specific topologies. Consid-ering today's state of the art, the primary requirements covered in this documentare for full-wave center tapped (FWCT) and full-wave bridge (FWBR) topolo-gies both feeding into shunt-capacitor filters. This material isnot to be used todesign poly-phase kilowatt power supplies, but rather, be used by those techni-cally competent, to Design low to medium power equipment for personal ornon-production intent to sell product or Design information issubject to local, state and federal laws and in most cases must meet otherregulatory 1 is a FWCT Rectifier , Figure 2 is a FWCT Rectifier with dual outputsand Figure 3 is a FWBR Rectifier .

5 Using conventional current flow,if is the totalforward current through the Rectifier diodes3,iO is the output current andiC isthe capacitor In this document, lower casev andi represent timeIntroduction to Rectifier DesignFigure center tap Rectifier andshunt-capacitor filter (Vi)++--VcRLFigure center tap Rectifier and shunt-capacitor filter with dual (Vi)vfRL2 ViViC1+-+-C2+-Vc1+-Vc2if1if2ic1io1ic2io2 1 Basic Electronics for Engineers and Scientists by Lueg and Reinhard 1972 International Textbook Company2 Radio Engineering by Frederick Terman, , 2nd Edition, McGraw-Hill Book Company, Inc. 19373 For FWBR this is the current through each diode, but for FWCT each diodes's current is times this displacement current - see Quantitative Analysis for further Electronics9108 FM 1972 Gilmer, TX 75645 February 20094 Rectifier Design and Analysisvariant values and upper caseIandV represent peak orrms values, noted also is Vrms is the transformer's secondaryrms voltage measuredfrom one end of the secondary to the center tap for FWCT or across the entiresecondary for 4 is an equivalent circuit applicable to both FWCT and FWBR rectifiersthat will be used as the basis of the calculations to follow.

6 The transformer willbe studied further, but for now it is assumed to have a turns ratio ofn =np/nswithnp andnsequal to the number of turns in the primary and secondarywindings. The transformer's leakage inductance isLLK and for the majority ofthis document is assumed to be The resistance of the transformer'swindings isRs', whereRss is the DC resistance of the secondary winding andRspis the DC resistance of the primary winding which couples into the secondaryat the ratio ofn2 soRs' =Rss + FWCT Rectifier with dual outputs, see Figure 5, is an exception to thecommon Analysis . Note each half of the center tap secondary has its' ownuniqueRss, assumingLLK is negligible, but the contribution of the primary,Rsp,is common to both outputs. If the two loads are always complementary this hasno significance; however, if the outputs can be widely different then it isimportant to note that the output drawing the larger current will reduce theoutput drawing the lesser current by the voltage drop acrossRsp/n2, that is to saythe two outputs are mutually coupled by the primary 5.

7 Full-wave center tap Rectifier and shunt-capacitor filter with dual outputs including commonprimary DC 4. Equivalent circuit for either FWBRor FWCT with appropriate Appendix F, Components, for additional 'nRdRwVfRcRLCVoVc+++ifVrmsicioViRs'nRdRw VfRcRL1C1+++if1 Vrmsic1io1 ViRcRL2C2+++if2 VrmsRs'nRdRwVfVo1Vo2Vc1Vc2 Rsp/n2ic2io2 Figure bridge Rectifier andshunt-capacitor filter +-vfvfC+-RLn(Vi)ViW5 BWC Electronics9108 FM 1972 Gilmer, TX 75645 February 20095 Rectifier Design and AnalysisCalculations for this topology are carried out for each output just as for aFWCT includingRsp/n2 in each calculation, as normal. When each output'svoltage and current values are known, then the effect of each output's currenton the other can be calculated by subtracting (Rsp/n2 )(io1) from output 2 voltageand visa transformer's intrinsic resistance prevents actual measurement ofVrmsunder load, so when monitoring the secondary with an oscilloscope it willappear the output is distorted.

8 In reality this is an artifact of the voltage dropcaused byRs as the pulsatingif is delivered to the load and filter capacitor. Thispulsating secondary current is the major difference in transformer serviceratings between resistive load and Rectifier contributor of series resistance is the forward dynamic resistance (Rd)of the Rectifier diodes. The diodes forward drop is subtracted from the peakvalue ofVrms in the calculations, but this value is static at the current level ofIO. Rectifier diode dynamic resistance is the change in voltage as a function ofa change in forward current (resistance). With modern silicon and Schottkydiodes this value is rather small, but none the less, a contributor and is addedtoRs' so thatRs =Rs' +nRd. +RwMost manufacturer's data sheets will provide a typical "Forward Characteris-tics" graph of forward voltage vs current and the slope of the curve is dynamicresistance, that isRd = final contributor toRs is the wiring resistance,Rw, which is comprised ofthe total resistance of the conductors interconnecting the transformer, rectifiersand filter capacitor.

9 For low current, high voltage power supplies this contrib-utor may be small, but for medium or high current, low voltage power suppliesthis is a significant Rectifier diode is represented byvf, and when on has the resistance andforward drop as stated and when off has no reverse current flow, a reasonableassumption with modern is assumed to have a significantESR ofRc, but insignificantESL, leakage current and dielectric absorption. For thecalculations to follow this is reasonable, using modern electrolytic capacitorsoperating at 50 or 60 will be covered further in discussingrms ripplecurrent, dielectric heating, and total ripple is assumed to be a fixed, pure resistance equal to the maximum steady stateload. If an electronic regulator is the load for the Rectifier system, then the loadis a constant current such thatiO(t) =IO that is to say the output current is nottime variant, but simply equal to the DC current, at least once steady-stateconditions are reached.

10 However, in this document the Rectifier load is as-1 See Appendix F, Components, for additional that in the following calculations and Figures the diode static forward drop is subtracted from the actual in-put voltage which then is identified Electronics9108 FM 1972 Gilmer, TX 75645 February 20096 Rectifier Design and Analysissumed resistive which has minimum effect on the results, except for outputripple current with a constant current load which, as just observed, is near zerobecause the regulator draws constant current over the range of the keeping with standard engineering practice, angular relationships withangles expressed in radians are presented in this document, in part to provideconsistency with the referenced published works and in part to simplify theanalysis. One other point of convention, in Figure 6 voltage and current areportrayed on a scale with a 0 starting point.