R-217 Design of Low-Power Pulse Transformers Using Ferrite Cores Nov52

8/2/2019 R-217 Design of Low-Power Pulse Transformers Using Ferrite Cores Nov52

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Report R-217

DESIGN OF LOW...POWER PULSE TRANSFORMERSUSING FERRITE CORES

by-Richard Dwight Robinson

DIGITAL COMPUTER LABORATORlMASSACHUSETTS INSTITUTE OF TECHNOLOGYCambridge 39, Massachusetts

November 3, 1952Thesis Date; August 29, 1952


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Report R-2l7FOREWORD

The recently developed group ot magnetic ferri ie materials presentI

attractive possibilities for use in the cores of transformers designed tohandle low-power short-duration voltage or current pulses. Because thesematerials possess high resistivity, the affect of eddT currents at highfrequencies is reduced. They may be used at higher effect!ve permeabili iesthan metals. Thus transformers with terri e cores can be physically verysmall, an ~ o r t a n t consideration 1n applications such as digital computerswhere large numbers ot transformers are required.

Because this thesis report, which has haa only, limited distribution,contains 1 n f ' o r m a t i ~ n on this subject of widespread interest, i t is peing is s ~ e d as a Digital Computer Laborator,y R-teries report.

"-a u ~ h o r is indebted to the staff and personnel ot the M.I.T.Digital Computer Laborator,y under the directorship ot Mr. W. Forresterto r presen.ting the thesis problem and assisting in this research. Specialacknowledgments are made to Mr. Robert E; HWlt of the Digital ComputerLaborator,y fo r developing the fabrication technique of the pulse transformersand to Mr. David J . Epstein of' the Laboratory for Insulation Research forconducting the d-c pulsetests on the ferrite materials.

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Report R-217ABSTRACT

This report presents the design procedure for constructing pulsetransformers of ferrite materials for Whirlwind I Digital Computer to replacethose made-or H)persil. The ferrite materials are ceramic-type materialswith the unusual combination of properties of ferromagnetism and high elect rical resistivity. The high res:lstivi y reduces the effect of eddy currentsat high frequencies 80 that higher effective permeabilities are observed thanwith metallic materials.

Effective pulse permeabilities of those ferrites tested range up toa maximum of 4$0 for O.l-microsecond pulses, while that observed for l-milHypersU (with air gap) is approximately 100.

Equivalent circuits for the 3:1 and the 1:1 pulse transformers inWWI when made from ferrite materials are developed. An analytical determinat ion of output waveshape is made using these simple equivalent circuits andshown to be ' in ' close accord with the observed experimental results.

The assemblY and circuit test of l-to-1 and 3-to-l ferrite transformers for use in Whirlwind I circuits are discussed. The transformersdescribed .re e q ~ i v a l e n t in performance to the Hypersil transformer nowused, and yet are more simply constructed and more compact, and can be produoed at considerably lower cost. Ferrite-core pulse transformers are admirably suited for O.l-microsecond pulse applications.

i i i


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Report R-217TABLil 0] ' CONTENTS

,FOUWORD Pa.ge

11

ABSTlU.Crr 111

CHAPl'ER 1 INTRODUCTION 1

CHAPlER II lvlAGNETIC TESTIIlG OF FERRITES 6A. 60-Cycle T es t s . . . . . .. . . . . .. . . . . . .. . . . . .. . . . . .. 6B. Pulse Te st s '.................................. 6C. Analysis of Pulse Loops 7D. Data'Summary 9E. Evaluation of :'&'erri tes for Pulse TransformerUse 11

.'CHAPlER I I I 14A. Introduction 14B. Basic Pulse Transformer Wave Shapes 15C. The 3:1 Pulse Transformer Equivalent Circu i t . 17D. A Design Procedure for 3:1 Pulse Transformers. 21

The 1:1 Pulse Transformer E q ~ i v a l e n t Circu i t . 26'". A Design Procedure for 1:1 Pulse Transformers. 30

GHAPrER IV PULSE T R A N S F O ~ ~ R CONSTRUCTION AND TEST 35A. Construction of Pulse Transformers 35B. Test Circuits for Pulse Transformers 36c. Test Results 37


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Report R-217

CHAPr.iR V

APmlIDIX

SUMMAJ;iY . . Page40

A. Apparatus for Taking Pulse Hysteresis Loops 45B. 3:1 PUlse Transformer Equations 47c. 1:1 Pulse Transformer Equations 52


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Report R-?17

Figure 1Figure' 2Figure .3Figure 4Figure 5Figure 6Figure 7Figure 6Figure 9Figure 10Figure 11Figure 12Figure 13Figure 14Figure 15Figure 16Figure 17Figure 18AFigure 18BFigure 19Figure 20

'rFigure 21Figure 22Figure 23Figure 24Figure 25Figure 26

LIST OF ILLUSTRATIONS

Drawing NumberA.'1976A.-'1975A-S2038A-52011A .$2012A-S2013A-,1969A-51974A-,2034A-52046A-52153A-52176A..521.32A-$21l0A-52131A - 5 2 1 6 ~ A-52201A-52205A""2244A-52227B-52006A-52007A-51978A-52136A-52093A-,2200A-52143


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Report R-217CBAPlIR IIftRODUCTIOB

A pulse transformer as defined by common usage is a transformerdesigned specifically for handling voltage or current pulses with timedurations in the order of micro- or millim1cro-seconds. Because these trans-formers perform only under high frequency excitations, the number of turnsrequired to produce a given voltage i s small, and for low-power applicationsthe physical size of a pulse transformer can be made correspondingly small.These transformers are common in radar and television circuitr,y, and inWhirlwind I (WI) computer over 3.000 pulse transformers are used for couplingbetween circuits where they permit a saving in the number and. size of tubesrequired.

The most common pulse shape which a pulse transformer in m isrequired to handle is a half-sine shape of 0.1 microsecond pulse duration.occurringat pulse repetition frequencies of from 1 to 2 mc. The two mostcommon voltage ratios are 3:1 an d 1:1. the former being used for mixing andremote point-to-point coupling, while the latter is used for local point-to-point coupling. For these applications Hypersil core transformers weredesigned by Wimettl an d have been in successful operation for several years.

The introduction of the new ferromagnetic ceramic materials or-ferri tesn on the market have caused several component manufacturers to intro-duce pulse tr.ansformers incorporating this material with resulting smallerphysical dimensions and l i g h ~ e r weight. A few samples of these ferri tetransformers were tested at Whirlwind, but found unsatisfactory. I t was

1. Wimett, T. E nLow Power Pulse Transformers.A Report R-122. Servomechanisms Laborator1. M. I . T December. 1947.-1 -


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R ~ p o r t R-217 -2 -

then decided that a research into pulse transformer design for Whirlwind 1using the ferr i te COTe materials' was warranted. This research would haveth e following objectives:

(a) To determine th e properties of those ferri tes commerciallyavailable as related to pulse transformer application.

(b) To for.mulate a design t e c h n i ~ e for ferri te pulse transformerswhich would give the particular performance required in WWI circuitr,y.

(c) To develop equivalent circuit of the completed pulse trans-former.

(d) To compare th e cost, performance and physical dimensions ofa ferri te core pulse transformer with the Hypersil transformers currently inuse in WI.

There were two reasons for th e f i r s t objective of determining theproperties of ferr i tes . Firs t , th e manufacturers of these materials inmost cases have published only d-c hysteresis loops and resi t ivi ty data.Second, in designing pulse transformers using metallic cores i t was necessaryto make assumptions concerning the magnetic state of the material due to theextreme eddy-current shielding effect 8 . Hence the core requirement s werestated only in great generalities. although attempts were made to define a

2pulse permeability." The high resist ivit ies of the ferri tes however givepromise of reducing the eddy-current shielding to such a low degree thatcorrelation of d-c data an d pulse response can be made. Such correlationwas obtained in this research and will be pointed out in this report.

The second objective, that of formulating a design technique,necessarily follows the introduction of any radically different material

2. Moody, lie F ., "A Treatise on th e De sign of Ptllse Transformers ForHandling Small Powers," THE 4310.


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Report R-217

in a component. For example . i t will be shown that in th e WWI 1:1 pulsetransformer a certain amount of leakage inductance is actually necessaryfor the desired performance. In the Hypersil design. this leakage wasreadily available, but in a ferri te design i t was necossary to introduce, i t by separating the windings.

One of the most useful tools.pf the electrical circuit designeris the equivalent circuit.. In the case of metallic-core pulse transformersan elaborate equivalent circuit has often become so complicated &s to losei t s value as an aid to the designer. The third ojbective then was to developa simple equivalent circuit which could be used to predict performance.Finding an equivalent circuit was successfully done in this thesis for twovery important reasons. Firs t . the ferr i te material by permitting a reductionin number of turns and in plysical size caused a corrosponding reduction indistributed capacitance and leakage inductance to a point where they nolonger became major parameters. Second, the shape of the pulse being considered was no t a square wave but a half-sine shape. For those who like tothink in terms of frequency response. the second consideration reduces thepass-band requirements of the transformer. thus reducing th e noticeableeffect of stray capacities.

The fourth and final objective of this research was to evaluatethe completed design. This is done in the f inal section of the thesis andi t should be noted that the new ferri tes find excellent application in pulsetransformers.

In the organization of this thesis report,the order of presentationis similar to the research objectives outlined above. In Ohapter I I , th einvestigations into the magnetic properties of some ferr i tes are described.and the resulting data analyzed to give criteria for optimum pulse transformer performance. InOhapter I I I , th e design of a 3:1 and a 1:1 pulse


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Report R-217transformer using the material properties of Chapter II are described.Before the actual design procedure is introduced, however, the equivalentcircuit is presented and an analytical determination of wave-shape responsemade. Chapter IV de.scribes the construction and test performance of thedesigns given in Chapter I I I . Finally, in Chapter V the thesis report issummarized.

I t shpuld be noted that the design procedures followed in thisthesis are for half-sine pulse shapes rather than the customary square wave.The analysis and design of pulse transformers for square-wave response arefair ly well developed in several texts and reportsJ and the reader i sreferred to these sources for the typical procedure. I f , however, th esquare-wave design procedures are followed for other pulse shapes, such asthe half-sine. one is faced with the problem of correlating the two per-.t0r.mances. I t is not sufficient to say that a pulse transformer which willpass a square wave will pass any wave shape. For while thi.s is true withregard to the positive-going portion of the pulse, one may. for example,desire a high over-shoot and fast decay on the negative-going portion ofthe pulse. Or as another example. the transformer may be required to "ring-when driven by a half-sine pulse. To correlate these required performanceswith square-wave response is difficul t i f the driving function i s no t asquare wave i tself . For this reason, the design procedure presented hereis for the particular case of a half-sine driving function.

Chapter II on ferri te core magnetic characteristics , is , however,applicable to any driving function design. Thus the values of pulse

J. For example: Wimett (Ref. 1), Moody (Ref. 2) and Glasoe and Lebacqz,Pulse Generators. M c G r a w ~ i l l Book Co Inc. Rad. Lab. Series No. S.pp. 499-660; or Lee, R., Electronic Transformers and Circuits, JohnWiley & Sons. Inc. , New York, 1947.


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Report R-217 -5-

permeability whioh were experimentally determined may be used for any pulsetransformer design where th e pulse width is in th e order of 0.1 microsecondsor greater.


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Report R-217CHAPl!ER II

MAGNETIC TESTING OF FERRITES

A variety of ferr i te materials in the form of small toroidswith dimensions 3/8 11 outside diameter. 3/16" inside diameter. and 1/8"thick were obtained from several ferri te manufacturers. These materialswere then given symmetrica160-oyole tests and high-and low-speed nonsymetrioal pulse loop tests .A. 60-CYGLI TEST

The main purpose of this tes t was to establish the general shapeof the low speed magnetization curve. For this purpose, the 60-cyclehysteresis-loop traoer in the Digital Computer Laborator,y was used. thisequipment having been constructed especially for the testing of small ferr i tecores. These curves were then checked with d-c hysteresis loops taken onth e Ciotti-type Bysteresograph in the Laborator,r for Insulation Research atM. 1. T. and the two methods were found to give i d e n t ~ c a l shape traces.B. RlYE TESTS

In order to perform high-speed pulse tests i t was necessar,r toconstruct some special equipment. This equipment which is shown in Figure 1and whioh is discussed in detai l in Appendix A oonsisted of a gas-tube pulsegenerator capable of producing half-sine pulses of current with time durationsof from 0.1 to 0.3 mioroseconds and with peak amplitudes up to 1 ampere. Thespeoimen to be tested was wound with 10 turns and placed in the c a t h o ~ e cirouit of the generator. The s t a n d a ~ prooedure followed was to integratethe voltage across the windings to measure flux density B and to read thevoltage across a small series resistor to measure magnetizing force H. Thisapparatus permitted the visual indication of a pulse hysteresis loop underconditions similar to those existing when the material is used as a pulsetransformer. -6-


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Report R-217 -7-

The presence of the Cioffi-type Hysteresograph at M. I . Toafforded the opportunity to draw, s l o w ~ t i m e - p u l s e hysteresis loops forcomparison with the high-speed loops o b t ~ i n e d from the ~ s - t u b e pulse genera-tor. For this test two concentric windings of 2S turns each were wound onthe specimen. The slow-t!me or nd_c pulse loops" were taken by startingwith the material in a virgin magnetic state an d applying a series of slowlyvarying unidirectional current excitations of constant peak amplitude.Following the procedure u s ~ d in the high-speed pulse tes t , the peak amplitudewas then reduced without cycling the material back to the ini t ia l virginstate and another d-c pulse loop drawn. This was repeated for several valuesof peak excita.tion.-c. ANALYSIS OF RJLSE LOOPS

The formation of a pulse hysteresis loop using the gas-tube pulsegenerator is shown in Figure 2. Because the technique of photographing theoscilloscope picture results in the time trace going from right to l e f t , thesketches wore'also drawn in this m ~ e r . Referring to Figure 2 i t is seenthat with Hl small. vl represents tbe e x c i t a ~ i o n current and v2 is the integral of T t , which is substantially th e voltage across the winding of thesample. The oscilloscope traces for each of these voltages are sketched inthe figure as well as the fOrmation of a pulse loop when VI is applied to thehorizontal deflection plates and v2 to the vertical deflection plates Of anoscilloscope. I t will be noted that following the completion of the currentpulse. the total voltage across th e winding vt does not return immediatelyto zero. a fact which is better i l lustrated by the slow decay of v2 ' the timeintegral of vto These sketches represent the observed results and will bediscussed next in conjunction with the formation of the slow time or d-c

These d-c pulse tests were conducted by Mr. D. J . Epstein of the Laboratoryfor Insulation Research, M. I . T.


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Report R-217 -B-

pulse loop.Figure J, "The Formation of a D-C Pulse Loop," is an actual trace

of th e pulse response of Ferramic H. taken on the Oioffi-type Hysteresograph.From the in i t ia l virgin condition the loci is seen to traverse to a Bma,x pointand then return to the residual induction point B . Succeeding pulses of thersame maximum H amplitude then tra.ce a pulse loop between Bmax a.nd Br Inthis kind of a slow trace, there is no problem of the material not respond-ing immediately to the current forcing function H. I f the speed of th e applieH were to be increased, however, a materia.l with a. low electrical resist ivi tywould have large induced eddy currents. T h ~ s e eddy currents would effectively"screen" much of the center portion of the test sample so that the outsidesurface would approach saturation while the inner portion remained relativelyundisturbed magnetically. I f the same va.lue of cross-sectional areas wereused for computing flux density B for the high-speed pulse traces as for theslow-speed traces, then for th e same amplitude of H there would be no similarin the values of the tota l 4 B ob served, ,( D. B = B - B ) .max r

1n the high resist-i vi ty ferri tes on the, other hand p for th e samemaximum applied H almost the same Ll:B was observed for the 0.1 microsecondpulse as for th e d-c pulse. This indicates that eddy currents are practically.negligible in these materials for frequencies up to 5 megacycles, th e frequenccorresponding to the half-sine pulse of width 0.1 microseconds.

In th e high-speed pulse loops for the ferri tes i t was observed thatwhile the end points B and B were the same as those for the d-c pulsemax rloop, the actual return path from B departed from that of the d-c pulse- maxloop. This i s shown in Figure J where the 0.1 microsecond pulse loop strikes

* The trade name of a ferri te produced by General Ceramics and Steat i teOorporation.


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Report R-217 -9-,a point:s I on theB-axis and then returns to B . From experiments i t wasr . . . r

found that as the maximum applied H was increased in th e fast-time pulsetests so that the material was driven further and further into saturation,the value of B I - B became a larger proportion of B ax - B . In fact ,r r m rfor the 0.1 microsecond pulses i t appears to be a good approximation to

that the return route from B i s along a straight line which has themaxsame slope as the init ial slope of the d-c return path.

The rate of time d e c ~ of the fast-time pulse loops from B I to Br .' rwas found to vary, depending upon the nature of the material under test .One material, MF-l13l.* which ~ s a rectangular d-c hysteresis loop ferr i te .has an extremely long decay rate as compared to the other ferri tes with non-rectangular d-c hysteresis loops. No explanation is given here for the factoraffecting this rate of decay other than to ,say that investigations were madeto assure that leakage capacitances, number of turns, and integration timeconstants were not causing this phenomenon.D. DATA SUltalARY

Figure 4 shows some typical 60-cycle hysteresis loops of the ferri tetested. I t will be noted that there i s available a fair ly wide choice in the

.d-c magnet.+c ~ h ~ r a c t e r 1 s t i c s . Figures Sand 6 show pulse hysteresis loops taken of several

materials at pulse lengths ranging from 0.1 to 0.3 microseconds. The valuesof dB ax andA H for the largest pulse loop is given under the figure.m maxThe voltage shapes below th e pulse loops are in the same order of presentationas those sketched in the to p right portion of Figure 2.

The'pulse magnetization response of various ferri tes is presented* A General Ceramics and Steatite Corporation material.


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Report R-217 -10-

in Figure 7 in the. form of a plot of AB vs. ~ B . These curves are loci ofth e tips of the high-speed pulse loops for various values of applied LlB .Included is a test result on l ~ m i l Bypersil. In computing the value offor Bypersil, the cross sectional area used does not include a laminationspace factor, so that th e AB values are slightly low.

Both the Bypersil and the Ferroxcube samples have air ~ p s . these being the only samples readily available for this research. The atfect. of the air gap on a magnetic material is to shear the d-c magnetization curve.This causes the residual induction Br to become less and permits a greaterrange of pulse operation before saturation is reached. By the proper amountof shear i t is also possible in some materials to realize a slightly greaterReffective pulse permeabilityR. lJ.e , which is here defined as l lB/ H. Thereason for this is that in shearing the hysteresis loop the pulse operationis further away from th e saturation regions and hence in the higher a-cpermeability regions.

Figure 8 shows the correlation between fast-time and d-c pulseresponse for Ferramic H. In this figure, the end-points 4Bmax' ~ H m a x ' areplotted for different values of pulse loop amplitudes. This shows thatwithin the limits of experimental error there is no difference in the endpoints of pulse loops for Ferramic B from pulse speeds of "d-c" to 0.1microsecond. (B' is no t concerned in this plot.)r

Figure 9 shows th e correlation between d-c pulse and 0.1 microsecondpulse response for various materials. Note the great discrepancy in the dataon Bypersil.

The significance of these test results i s that for pulse lengthsat least as short as 0.1 microsecond the d-c magnetization curve may be used4. M. 1 . T. Staff, Magnetic Circuits and T r ~ s f o r m e r s . pp. 65-68, John Wiley

& Sons, 1943.


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Report R ~ 1 2 7 -11-

to predict the performance of the high-resistivity ferri tes to ~ s e excitations. This may be done by considering th e material is init ially at theresidual induction value Br of Figure J (a s i t certainly will be after thef i rs t pulse). Then a d-c pulse loop of the general shape shown in Figure Jmay be sketched. The resulting line from.B out to Ba'V' may then serve asr mg"n.loci for L:\B and .a H values for use in fast pulse calculations.

Table 1 summarizes th e data for the materials tested, and includesth e information that was available from the manufacturers. The discrepancyin the values of B between th e Test Data and ,the Manufacturer's Data is 'max .due to the difference in applied Hmax for the two cases.B .' EVALUATION OF FERRlTES FOR PULSE TRANSFORMERS

From Table 1 i t can be seen that the effective pulse permeabilitiesof those ferri tes tested range up to a maximum of 450. For 0.1 microsecondpulses this is a much better value than for grain-oriented steels. 2Moodygives values of ~ J O for J-milHypersil and 210 for 4-mil Mumetal at this pulselength. For 1 miorosecond pulses these values i n c ~ e a . s e to 400 and 720 respec-t ively. Thus on the basis of pulse permeability alone, i t 1s seen that theferri te is a superior material for pulse lengths shorter than about 0.5 micro-seconds.

The ferri tes lose some of their advantage by having relatively lowsaturation inductions. This means that i f the material is used for long pulseor for high voltages then th e core area must be increased sufficiently toprevent the material from going into saturation. As was m e ~ t i o n e d earlier i tis advantageous to introduce an a ir gap in these cases to permit higher valuesof permissible AB.

Another disadvantage of th e ferri tes is their low Curie temperaturei . e . - the temperature at whioh the material loses i t s magnetic properties.

'. This may be serious i f the transformer is to be mounted near a source of heat.

2. Ibid.


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Report R-217 -12-

and the designer may well keep this in mind when laying out.his circuit .For the application in Whirlwind I , none of these disadvantages

are particularly important. The 3:1 and 1:1 pulse transformers are low-power devices and operate under 0.1 microsecond pulse excitations. Furthermore, there is no particular problem of heat dissipation since the circuitsare well laid out for ventilation. Thus, from the standpoint of the magneticproperties the' ferri tes would appear to f i t this application well. The nextchapter will deal with the problem of designing t ~ e 3:1 and 1:1 ferrite-corepulse transformers for WWI.


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Material

Ceramag 7ACeramag 5N

Ferranlic H

Ferroxcube )C(witha1r gap)

Ferricore)'1F-1l31

Manufacturers' Data Test Da.ta(at H = 7 oersteds)maxf(ohm-cm) Curie Maximum ValuesTemp. BrGauss3 x 105 165C 3600

1 x 104 150C 1470

1 x 102 1600 1200

H B B H B 6,Bmax average effectivec max r c max e pulse permeabilityOersteds Gauss Gauss Oersteds Gauss0.6 4160 1100 0018 2380 1280 350

1150 0.45 2510 1)60 3800.18 )400 1600 0.18 2320 720 4500. 2 )500 450

950 1.06 2300 1350 2301900 1.0) 2;00 400 40

)1a.teria1Stackpole Carbon Company

" " nCeramag 7ACeramag 5NFerl"amic HMF-1131Ferroxcube ]CFerr1core

General Ceramics & Steat i te Corp_

TABLE I

" nan nFerroxcube Corp_ of AmericaFerricore Company

SUJ.DiARY OF 1-1AGNETIC DATA ON FERRlTES

-l.J-


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Report R-217CHAP.l!U I I I

RJLSE TRAliSFOBMER DESIGli

A. INTRODUCTIONIn designing pulse transformers-for IVI circuitry, there are

several objectives that the designer must keep in'mind. As an example,for the 3:1 pulse transformer which operates under 0.1 microsecond half-sineexcitation, these objectives include wave shape preservation of the positiveportion the pulse and rapid recovery of the overshoot. Before the designcan be attempted then, the designer must understand the effects of the varioustransformer parameters on th e output wave shape. These parameters, which in-elude mutual inductance. leakage inductance, distributed capacitance and anequlTalent 108s resistance, will assume various degrees of importance inshaping the output waveform. J'urthermore, the kind. of loading and th e nature

I

of the driving souroe will a180 i n f l u ~ n c e the finkl shape of th e transformedpulse ,so that i t 1s neoessary to consider the entire circuit in which thetransformer is to operate in the design procedure.

For these r e a s o ~ s . a oonsiderable portion of this chapter is spenton showing the distortions produced by the different circuit parameters. Todevelop this Afeeling for the parameter effects,a "Basic Transformer" 1sf i rst introduced which has neither leakage inductance nor distributed capaci-tance. The magnetizing current and the output volta.ge shapes of such a trans-former are described when driven by a half-sine pulse current source for thecondi t ions of no-load. and of large load. ,Constant reference is made to theB-H plane in order to relate the magnetic state of the core material to outputwave shape.

Following these concepts of th e Basic Trans.former. the equiva.lentcircuit for a 3:1 pulse transformer to be used in IVI applica.tions is presented

-14-


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Report R-217 -15-

The analytical determination of output voltage wave shape is carried ou t andthe design procedure described. This in turn is followed by the equivalentcircuit aDd design procedure for a 1:1 pulse transformer when operated underno-load conditions, this being the condition of operation in WW1 of the 1:1transformer.B. BASIC RJLSE TRAliSFORJilER WAVB SHAPES

The pulse loops of Figures 5 and 6 were taken with a half-sinewave of current excitation. This is essentially the condition of a pulsetransformer under open-circuit conditions when driven by a current source suchas a pentode in Class B operation with a half-sine wave impressed upon thegrid. Neglecting the practical considerations of leakage flux, stray andwinding capacitances, the transformer would produce across i t s output terminalsa symetrical cosine voltage shape as i l lustrated in the portion of Figure 10where a constant permeability is assumed.

I f , however, the permeability were to change at the peak of themagnetizing current wave form to some smaller value , then the negativepeak of the voltage wave vt would have a smaller absolute magnitude than thepositive peak. Assuming a. quasi-steady state condition, after time T thematerial must return from B to i t s ini tal starting point B Sincer , r

t = T/2 P2 t ~ C I ( ) ft dt = NS d/J = -~ ~ t dtt =0 P t = T/2the areas under the positive and negative portions of the voltage plot mustbe equal, or th e integral v2 = ~ V t dt must go to zero. This decay of v2was described in the preceeding chapter a.s dependent upon an intrinsic lagof the material.


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Report R-217 . -16-

Bext consider the effect of placing a load resistor across thetransformer terminals as in Figure 11 . I f R is sufficiently small so thatth e magnetizing current through the transformer is bu t a fraction of thecurrent through R, the half-sine current source may be replaced with a half-sine voltage source with series resistance R. The current i in this equivamlent circuit is th e magnetizing current through the basic transformer.

Because im is proportional to th e integral 'o f vt i t will beapproximately a cosine-shape wave from time t = 0 to t = T. I t is interestingto note that from t = 0 to t = T i is always increasing so that the pulsemhysteresis loop is now traversed from B B Oy in twice the time as withr mQIAthe open-circuited transformer. With the material in a magnetic state ofHmax' Bmax th e voltage source is shorted and th e transformer is lef t with th eload resistor R across i ts terminals. Since the magnetizing current is s t i l lf ini te , i t continues to flow and causes a reverse peak on v t of magnitude 1m R.For the case of constant permeability, the reverse voltage of vt then goes tozero with an exponential decay of time constant L/R. That this sat isfies thecondition of equal areas under the positive and negative portions of the vt wavcan be shown by let t ing the voltage vt ?e a true half-sine wave from t = 0 tot =T. Then

For 0 t T

at t = T.

im = 1m =Sl s in wt d(wt) = o.!uiLo51Tet d t = -o./wo

Thus the areas are equal.

e 'tFor t> TIRe -Rt /Lm

as t "

00dt = -2ER r e-Rt/ L dtwL Jo

- - 2J/w


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Report R-217 -17-I f the case of a change in permeability is considered as in

Figure 11, then the shape of th e decay of the negative overshoot in vt willno longer be a single exponential curve. In terms of the pulse loops ofFigure ) . for R very small the d-c pulse return path will be followed. AsR inoreases. the loci will approach that of the open-circuited case discussedpreviously. In any case, the in i t ia l slope of the returning loci will bethat of the d-c return path. In Figure 11, the change in permeability ispiotured in distinot steps which result in the three permeability valuesJ.J., p.1 , and J,J.' ' . The effect s of J,J.' and JJ." on the voltage overshoot are togiye a two different time constant exponential decay. Compared with the oon-stant J,J. case, the, overshoot will at f irs t decay more rapidly, and then attime T' will follow a very much slower rate of decay. Again, however, th eareas of the positive and negative portions of the vt curve must be equal.

From this consideration of the effect of load on the output waveshape of the basic transformer. th e next section describes the ) :1 pulsetransformer e,quivslent circuit . and. i t s analysis.c. THE J: 1 PULSE TRANSFOmlER ECPIVALEUT ClltCUIT

A standard equivalent circuit which is generally drawn for a pulsetransformer is the general equivalent circuit of FigUre 12. In this ci rcui t .a l l elements are referred to the primary. Here Rl and R2 represent windingresistances of the primary and secondary windings, R represents the transeformer losses, J 1 and 12 the leakage inductanoes, L the primary self in-ductanoe or "open circuit" inductance, C the winding-to-winding capacitance,mand C and C include the distributed cauacitance of the windings as a lumpedp s -parameter along with the external stray. source, and load capacitances.

Because this circuit i s too complicated for analysis, i t is commonpractice to neglect those parameters which produce second-order effects and

...to resort to a reduced equivalent circuit . The l e a ~ g ~ inductance,f l and '


24/87

Report R-211 -18-

a 2 ~ 2 are lumped into a single element ,R t the winding resistances areneglected, and the effects of C combined with 0 when the voltage ratio am pi s greater than 1 , or with 0 i f a is less than 1 . The equivalent transsformer loss resistor R i s also omitted.e

In measurement s taken on the 3: 1 pulse transformers design"ed inthis research, i t was found that the leakage inductance was so small that i tcould be n e g l e c t ~ d for the analysis of the response of the transformer underload conditions. For this reason, i t was omitted in th e f inal diagram ofFigure 12. The capacitances were then lumped into a single Ot and the resis-tances combined into a.single R. To show the effects upon output voltagewave shape of each of the parameters L, R, and C, the basic transformer isagain introduced in Figure 13.

Previously i t had been assumed that the magnetizing current wasnegligible so that true half-sine of voltage appeared across the transformerterminals. In Figure 13, the magnetizing current is no longer considerednegligible in comparison with the current through R. These curves were plottefrom equations (1), ( la) , (2 ) and (2a) in Appendix B. Part 1, for a half-sinecurrent source of pulse width 0.1 microseconds. These plots are for typicalvalues of magnetizing inductance when referred to the secondary of a 3:1 pulsetransformer with a load resistance of 93 ohms. For a given maximum amplitudeof half-sine current excitation,the effect of reducing the magnetizing in-ductance ia: (1 ) to reduce the amplitude of the positive-going output voltage(2) to shorten the pulse length, and (3) to increase the amplitude of over-, .. "shoot. The recovery time of the overshoot is lessened, however, because thein i t ia l overshoot amplitude is greater and because the time constant L/R isless. This i s an important consideration when the pulse repetition frequencyIbecomes high.


25/87

Report R-217 -19-Figure 14 i l lustrates the effect of capacitance C upon the output

wave shape and magnetizing current. Equations (J). r3a) , (4) and (4a) ofAppendix B, Part 2, were used to plot these curves. The curves for C = 0are curves 2 of Figure 13. and are used for reference because they have thesame .values of R and L. The main effect of capacitance C is to delay the

...

rise of t h ~ pulse as can be seen from physical reasoning. This has theapparent effect of stretching the pulse width an d rounding off the sharpcorners of the overshoot peak.

Thus far, the inductance of the transformer has been assumed con-stant and the analysis confined to a l inear network. In Figure 15. thepermeability of the material is permi,tted to change at th e peak of the magnet-izing current wave shape as was done in Figure 11 for th e Basic Transformer.Equations (5) an d (6) of Appendix B, Part 3. were used to calculate theresponse after time wt =n. At this time, th e new value of permeability

was a s ~ e d to change to one-half th e value o f ~ . Actually the magnetiz-ing current peak occurs slightly before wt = n. but for simplicity in calculation this peak assumed at wt =n. (For large C, this error is slight. )

At some later time when the m ~ ~ e t i z i n g current i has decreasedmto less than one-third i t s peak value, the permeability again changes to amuch greater value ~ I I . In Figure 15. this change is assumed at wt = 2". andwhile no calculations were made. the curves are sketched to show a new long-time-constant beyond this point.

The effect of this change in permeability is to cause a "bump" onth e overshoot voltage that causes an apparent rapid recovery. However, sincethe equal area condition must s t i l l be fullfilled. the curve for constant J..tsoon crosses th e overshoot voltage curve for variable

Figure 15 represents an analytical prediction of the output voltagewave shape of 3:1 pulse transformer with a Ferramic H core designed to replace


26/87

Report R-217 -20-th e WWI Hypersil-core transformer. The actual response of this transformeras viewed on an oscilloscope i s shown in Figure 23, 1193 Ohm Load ," and will bediscussed la ter in Chapter IV. I t may be mentioned here, however, that cal-culated response using the equivalent circuit in Figure 15 gives good correIa-tion ,\fi th observed response.

,. . Before proceeding to the next section which deals with th e actual3:1 transformer design, the effects of pulse repetition frequency upon voltagewave shape will be discussed. Consider what happens when a second pulsefollows so close to the f i r s t pulse that i has not yet decayed to zero. Inmterms of the d-e pulse hysteresis loop of Figure :3 (consider R small),thismeans that in the B-H plane the next pulse loop will no t s tar t at B , bu t rathrat some point s t i l l on the return path from Bmaxo The new pul.se loop willthen trace out to a region of greater saturation, which means a lower perme-abili ty. Not only will th e second voltage pulse amplitude be reduced becauseof th e lower permeability; but since i t began before the overshoot of the f i r spulse had fully decayed. there will be a downward shif t of the base l ine . Along sequence of such pulses will cause the material to eventually reach aquasi-steady s t ~ t e condition somewhere in the f irs t quadrant in th e B-H planewhere the resulting magnetizing inductance is low enough and the overshoots

(high enough (refer Figure 13) as to cause the areas of the positive portionsof the pulses to be equal to the areas of the negative overshootso I f theload resistance were very high so that the near-open-circuited hysteresispulse loops were executed, the situation could be aggravated since th e secondpulse loop would begin almost at B ' of Figure 3. However, this would requirerthe pulses to be very close together since the large R condition gives highovershoots and fast decays. In some very special ca$es where this situationarises, th e magnetization lag mentioned in Chapter II may have to be con-sidered.


27/87

Report R-217

The extent of voltage amplitude reduction and base line shift isthus a function of the pulse repetition frequency, the in i t ia l overshootmagnitude, the overshoot decay rate, and the degree of in i t ia l saturation inthe material -- i .e . , how far up the B-H plane the material can go before saturation. I f the material is operated over only a small portion of i t s permissible 6. B range; and i f only a. few pulses occur in the pulse chain, then theremay no t be a diminution in absolute amplitude 0+ the voltage pulses bu t merelya shift in the base l ine.D. DESIGU P R O C E D U ~ ]'OR J: 1 RJLSE TRANS1fORIvJERS

I t has been shown that there are several factors to be consideredin preparing for a pulse transformer design. These include,.the nature of thedriving source, the nature of the load, the desired pulse length and conf i ~ l r a t i o n , and the pulse repetition frequency. To these factors can be addedthe output voltage amplitude desired, the turns ra t io, and the core material .The practical considerat.ions of a suitable winding procedure and the coremount 'will also influence the design where large quantities are concerned.

Because the external circuit parameters have such &1 importantbearing upon the observHd output wa.ve sha:qe of a pulse transformer, i t isgenerally a more pra.ctical procAdure to be guided by a fe\'1 f1Uldaraental relationships and obtain the optimum performance by experimentation rather thanto attempt a complete design on paper before an experiment is made. The reasofor r ( ~ l y i n g , upon experimenta.tion becomes obvious when the designer attemptsto calculate leakage inductance and self capacitance of a pulse transformer.Il t best, these calculations can give only orders of magnitude for the smallnumbers of turns which are involved here. Thus i t becomes necessary toconstruct a few sarnple transformers and measure these unknown. quantities.These measurements together with the observed performance of the pulse transformer in a test circuit will then serve to guide the direction for optimum


28/87

Report R-217 -22-

perf ormance.I f the transformer i s to be used in a variety of circuits, th e

designer must arbitrarily select a typical circuit as his test of performanceand base his design upon i t . Other circuits may then require minor modifica-tions to adapt to this design.

In th e case of the 3:1 design for mil application, the problemwas to develop a ferrite-core, pulse transformer which could replace anexisting H y p e r ~ i l - c o r e model. The problem was well defined in that thecircuit applications were known and the existing Hypersil transforLler couldbe w?ed as a standard for performance measurement. To i l lustrate a procedurewhich may be followed under such circumstances, the actual design of an accept-able ferrite-core 3:1 pulse transformer is carried out.

To simulate the conditions of operation, a Basic Point-To-PointAmplifier C,ircuit as sho",n in Figure ~ ~ O was selected. j'rom this circuit . andfrom t h ~ . general operating requirements in WWI, the following specifications

-

for th e design were established:1. The pulse shape is 0.1 microsecond, half-sine.2. The maximum pulse repetition frequency is 2 megacycles.

\3. The output pulse amplitude is to have a maximum value' of 20volts peak, the transformer being 3:1 stepdown.

4. The driving source is a power pentode which may be consideredas a current source.

5. The load resistance can be considered as 93 ohms, referred tothe seconda.ry.

6. The tota.l circuit capacitances. exclusive of th e tr,911sformer,are 10 Dllllf in the prim,ary, and 50 j',unf ll} the secondary.

l"'lrom these specifications, the pulse transformer was to fu l f i l lthese requirements:


29/87

Report R-217 -23-1. Pass the half-sine pulse with as l i t t le distortion as possible.2. To have as small a pulse-repeti tion-frequency sensi t iv i ty as

possible--i .e. , to have as l i t t le diminution in amplitude and shifting ofbase l ine for a train of pulses a.s possible.

3. To be of small size and ea.sily mountable.4. To be capable of m a . n u f ~ c t u r e a t a reasonable cost.For a core shape, a small toroid of d i m e n s i o n s ~ p r o x i m a t e l y 3/8"

out side diameter, 3/16" inside diameter and. 1/811 thick was selected. Thematerial used was Ferramic H. I t was decided to base the design upon thepulse-repetition-frequency sensitivity requirement. This 1s related to themagnetizing inductance L of the transformer in the following development,using the equivalent circuit of Figure B-1. Appendix B.

Let T =half-sine pulse width =0.1 microsecondT' = time between pulses in seconds = ~ 0 . 4 microseconds

= time constant of recovery voltage =L/R.In a time period of 4 time constants, a simple exponential will have

decayed to less than 2% of zero.Let "1'; = T' /4 = 0.1 microsecondsSince R = 93 ohms.

L = !r)'R = 9.3 microhenriesl) referred to the secondary.From eQuation (7) , Appendix B, Part 4 .

J ~ o 8 -He = 0 . 4 ~ = number of secondary turns.eFrom the core measurements and the value of = 450 (Table 1):e1 = 2.24 cmm


30/87

Report R-217

So that

-24-

903 x 2.24 x 1020.4". 0.076 x 450Using N =7. lip = 3 x 7 = 21 prilllary turns.Reca.lculating the inductance L,

L = 904 microhenries. referred to the secondary.ill -1 R -1 93 6Since T = tan wL = tan 2TT10+7 (9.4)10- = 17-1/20 ,

curve 2 of Figure 13 applies, with capacitance and non-linearity not yetconsidered.

An approximation of the maximum flux density, Bmax' for the operating conditions can be made using Formula (8) of Appendix B, Part 4 ,

ABmax

.oBmax

20 volts,

This figure will be high. because th e a.ctual voltage shape across thtransformer requires less flux than a 0.1 microsecond half-sine.

From Table 1 or Figure 8, i t is seen that the material will be wellbelow saturation.

There is no accurate method of calculating the distributed c a p a c 1 t a nand leakage inductance of this shape of toroid with so few turns, so thesevalues were measured after th e core was w o ~ with 21 and 7 turns of No. 36Double Formex Wire. To keep th e leakage inductance small, the primary andsecondary were concentrically wound.

Measurements on a Q-meter gave the following readings at 5 mcs.em = winding-to-winding capacitance = 8 mmf.


31/87

Report R-217 -25-

0p = primary distributed capacitance = 2 mmf.= leakage inductance (short circuit inductance), too small

to be measured.Referring al l the capacitances to th e secondary:

o = 9(0 + 0 ) + 0 + C =9(2 + 10) +8 +50 = 166 mmf.p source m loadA check is made to insure that the roots of the characteristic

equation of the R,C ,L parallel combination are real (see Appendix B, Part 2).1(2RC)2 = 1(2x93x166xlO-12)2 =

11to = 9.4xl66xlO-18 14= 6.4 x 101 1Since (2RO)2 LO ' the overshoot will be overdamped, which is the

situation desired.An analytical plot of the output wave shape for the transformer

parameters will give a curve similar to t h ~ s e of Figure 14. Because a curvefor 0 = 166 mmf is not shown, th e curve for 0 = 90 mmf will be used to i l lus-t rate the calculation of B (for the c a l c u l ~ t i o n s below, this value ismaxnot much in error.)

From Figure 14 for C = 90 mmf, the maximum positive voltage etm = 72Also. 1m = maximum im = .42 I . Hence, 1m = .0058 e tm , referred to th e seconda

For et = 20 v , I = (.0058 x 20) = .116 amps.m m-IImax I-e = 0.4nNlm J.L _ O,4TI7(.116) (450) = 205 g a u s ~ . 1 e - 2.24m

To produce 20 volts across the 93 ohm load resistor . the currentsource must be capable of providing a peak current of:

I = 20/72 = .278 amperes, referred to the secondary,or

lp = .278 x 1/3 = 92 ma in th e primary.


32/87

Report R-217 -26-

The effect of non-linearity in L will produce a change in theshape of the overshoot similar to that shown in Figure 15. For this lowvalue of i1B x however, the change in permeability will not be as great asmath e 2:1 change pictured in Figure 15 .

For computing values of Bmax and current source r e q ~ i r e m e n t s , l i t t le error i s introduced by using the curves of Figure 13. provided thecapacitive loading is not excessive.

The actual construction and circuit operation of this transformerdesign will be discussed in Chapter IV. I t should be noted from curve 2 ofFigure 13. however. that the magnetizing current near th e end of the pulserises to a value which is over 40% of the peak supply current. For squarepulses. the usual criterion is a maximum magnetizing current of 10% of thepeak supply current. Applying this condition to the half-sine pulse wouldresult in curve 5 of Figure 13 which has a small amplitude overshoot with along-time decay.

The next section will deal with the design of a 1:1 pulse trans-former for WWI circuitry. In this case, the transformer is operated underopen-circuit conditions and is allowed to "ring." Because the leakageinductance plays such an important part in determining th e ringing character-is t ic , a different equivalent circuit i s used for the 1:1 design than th esimple L,R,C parallel combination used in the 3:1 design.:m 0 THBl 1: 1 RJLSE TRANSFORMER EQUIV LEN"T CROUIT

The most common application for 1:1 pulse t r a n s f o ~ e r s in WWI isfor local point-to-point coupling between two amplifiers adjacently located.Here the secondary of th e transformer i s coupled directly to the grid of theadjacent amplifier with no load termination. This gives a high amplitude tothe positive pulse as the transformer parameters combine with the externalcircuit constants to produce a resonant "ringing" effecto (See Figure 26)


33/87

Report R-217 -27-

Because only the f i r s t positive pulse i s desired!) a diode is connectedacross th e transformer so that on the negative swing a low resistance loadis presento This causes the circuit to suddenly become overdamped on th enegative voltage swing an d the energy of the circuit i s dissipated in the diodeforward resistanceo (Actually a small resistance is included in series withthe diode in order to produce th e desired amount of dampingo)

In applying the 1:1 Hypersil-core pulse transformer in the mannerdescribed above!) i t was found that in spite of the damping diode an externalshunt inductance of about. 50 microhenries was necessary across the transformerterminals in order to prevent the recurrence of positive pulseso An analysisof the ci rcui t with th e external shunt inductance was diff icul t due to th enon-linear behavior of the Hypersil core at these high frequencieso Inferr i tes 9 however!) the eddy currents are very small and i t was found that anequivalent circuit could be developed which would not only show the oscil la-t ion frequencies involved but would also explain the action of the externalshunt inductanceo This equivalent circuit will now. be discussedo

Because th e leakage i n d u c t a n c ~ p l ~ s such' an important part indetermining the ringing frequencies of the 1:1 pulse transformer 9 i t isnecessary to include i t in the circuit developmento Consider the 1:1. trans-former as a coupled circuit as shown at the top of Figure 160 Here Llrepresents the primary self-inductance and M the mutual inductance between th etwo circuitso Because this is a 1:1 'transformer 9 the secondary s e ~ f - i n d u c t a n c eis numerically equal to Ll0 C1 and C2 represent th e sum of a l l the capacitanceon the primary and secondary respectivelyo

M2Defining a '!leakage coefficient" cr = (1 -L2 ) 9 the transformerI

equivalent circuit can be drawn as shown in th e second diagram of Figure 1605

50 Guillemin D Eo Ao 9 Communication Networks!) Volo 10 ppo ,310-,32,3 I) John Wiley &Sons ) 19,310


34/87

Report R-217 -28-

I f the re sistances a.re then omitted. the third diagram results where )..,=sum of .the prima.ryand secondary leakage 1 n d u c t a n c ~ s * 1.1a-.. andL =magnetiz ..' i ~ inductance = L1(1-0") . , .

For clari ty the current source I is now replaced with a voltagesource V = -ff-1 [1 dt . and th e equivalent circuit for the 1: 1 pulse transformer with no load 1s complete.

Inspection of this new circuit shows that i t consists of a voltagesource supplying a series resonant circuit ... 01 and a parallel "anti-resonant"

. circuit .1,.2 - The resonant and ant.i-resonant frequencies are defined as f l(or WI) .and. f 2 (or W2) re spect vely.

Figure 17 i l lustrates the use of susceptance plots for determiningth e natural behavior of the transformer equivalent circuit . The ratio ofoutput voltage E2 to driving source voltage V or "transmissionn T(jW) 1sequal to the admittance of the series circuit 11 divided by th e sum of thea d m i t t ~ ~ 7 s of t h ~ series and paraliel circuits I1 + Y2. The poles (infini t ies) of T(jw). which are the oscillation frequencies of the n e t w o r ~ , aredetermined by finding the zeros of the sam of the admittances Y1 + Y2- Thismay be done graphically by adding the two separate admittance plots for Ylarid Y2 as shown in Figure 17. The natural frequencies or oscillation frequencies of the network are then defined a s ~ ' a n d ~ ' I . Note that wl an dw2 l ie between the natural frequencies of the network.

In Appendix 0. Part 1. these oscillation frequencies are shown to beequal to: tlJi" =jd. +/ -2 ..,.

tlJi' Vf.-jd-2 -pwhere, 20< = '"\'tc + J l ~ \..) 2 1

l12)( l J )

(10)

(II)


35/87

Report R-217 -29-I f the transformer is now driven from a. current source having a hal t-

sine pulse shape, the output voltage 2 may be calculated. Thetransientresponse to this excitation is calculated in AppendixC , Pa r t 2, and the re-sults are given by equations (14) and 18) for the periods during t ~ e dr.ivings o u ~ c e pulse length and after the driving sOurce pulse termination respectively

ForO ' -wt n.

FQr 1T wt,

Where 1K = 00 oW;1 2t

x= -Wy =

W

(15)

(16)

In these expressions I is the maximum amplitude of th e current pulsean d w is the fundamental frequency component of the driving 8 0 u r C ~ t i . e . , for0.1 microsecond half-sine pulse w = 5 megacycles.

Plots of the transient response of the 1:1 pulse transformer equiva-lent circuit for various values of x and yare given in FigurelBA. Thesecurves may be used to point th e direction for proper ~ a l u e s of x and y to give

I

a desired response. An important point to be observed is the pulse delay in-troduced when the pulse transformer is operated under t h e s e o p e n ~ c i r c u i t conditions (compare the curves of Figure lBA with those of Figure 13).

The type of analysis which has been presented here will be used inthe next section to predict the behavior of a ferrite-core 1:1 pulse trans ...former which was designed to replace th e standa.rd WW1 Hypersil-core transformer


36/87

Report R-217 -30-

The effect of adding an external shunt inductance or "tail-reversing inductance"will be included in the analysis.F. DESIGN PROCEDUBI FOR 1: 1 lULSE TRANSl!'OBMERS

As in the case of the 3:1 design. the 1:1 ferrite-core pulse trans-former was designed to replace an existing model--in this case the WWI standard1:1 transformer with a Hypersil core. In the original design the Hypersil

1pulse transformer was specified to work into a load of 1,000 ohms. However.i t was found that a higher amplitude pulse could be obtained, by operating thetransformer open-circui ted. Because th e resultant ,.ringing occurred partiallyabove the base l ine so that a damping diode was not effective in preventingth e recurrence of a positive p u ~ s e . (see Figure 26 "51 Open S Closed") t anexternal shunt inductance called rlTail Reversing Inductance" was added. Theeffect of this inductance i s to reduce the L of Figure 17 so that W2 andhence WR' are increased. Thus after th e f irs t pulse the ringing occurs belowthe voltage reference line where the damping diode is effective in "clipping"th e pulse.

Following the procedure for the 3:1. a test circuit was bui l t whichwould simulate a typical operating condition. This circuit is shown at thetop of Figure 20 and will be discussed in more detai l in the next chapter.Because the relative importance of the pulse transformer parameters had neverbeen developed for this application, i t was necessary to use an experimentalapproach to match the performance of the ferrite-core transformer with thatof the Hypersil transformer. In doing this . i t soon became apparent thatthe existance of a f ini te amount of leakage inductance was essential to thetransformer's operation. Because a toroidal shape ha d been selected forpurposes of standardization with the 3:1 design, i t was necessary to separatethe windings to introduce this leakage. Based on the observed performance1. Ibid.


37/87

Report R-217 -31-

',0:(, th e Hypersil 1:1 transformer, the requirements set forth for th e 1:1 ferri tedesign were then as follows:

1. The driving force i s a half-sine current pulse of 0.1 micro-second duration.

2. The maximum pulse repeti t'ion frequency is 2 megacycles.3. For the test circuit of Figure 20. th e output pulse amplitude

i s to be 20 volts peak.4. The combination of a damping diode-and-resistor together with

a 50 microhenry t a i l reversing inductance must prevent the recurrence ofanother positive swing after the in i t ia l positive pulse.

5. For purposes of standardization, the geometrical shape of the1:1 transformer and th e 3:1 transformers are to be the same.

In the case of the 1:1 transformers, the pulse-repetition frequencysensitivity was not to be the basis of design since the overshoot is a doublefrequency transient which cannot be resolved into a simple exponential decay.Instead, the two-humped overshoot of the Hypersil transformer was duplicatedby experimentation with a ferrite-core transformer. (see Figure 26, "Sl andS Closed").

This ferrite-core transformer is a Ferramic H toroid of th e samedimensions as the 3:1, and has two windings of 24 turns of No. 36 DoubleFormex wire wound on opposite quadrants of the toroidal shape. From measure-ments on a Q-meter. the transformer capacitances and leakage inductance werefound to be: c = c =5 mmf.P s

Lle:= 15 microhenries.From formula (7) of Appendix B, Part 4, for 24 turns, and = 450L = 0 . 4 T I N 2 ~ 1 0 - 8 0.4 n(24)2(O.078) (450)10-8 = 110 microhenries1 1m (2.24)Thus q-= 15 - 0.13.110


38/87

Report R-217 -32-

From measurements taken on the test circuit (including oscilloscopeprobe), primary capacitance exclusive of the transformer = 18 mmf, andsecondary capacitance (without tail-reversing inductance) = 52 mmf.

Using the equivalent circuit developed in Figure 16 and th e mathe-matical expressions for output voltage derived in the last section, theresponse of this transformer was calculated for the conditions without th eexternal shunt inductance and with th e external inductance. Without theinductance, the following p a r a m ~ t e r values apply to Figure 16:

L =Ll(l-cr) =110 -15 = microhenries. 15 microhenries

C1 =5, + 18 = 23 mm!.C2 =5 + 52 = 57 mmf.From the relationships derived in Appendix C, Part 1,

1 . 7wl =JJrCi = 5.4 x 101 . 7w2 LC

2= 1.36 x 10

2 2(j . =.2!,f,-+ w1 = 2016 x 10152 './= w2 w2 = 53.6 x"10281 2

UJt' ,eJ ( +/0


39/87

Report R-217 -33-.so that the voltage va.lue at &nl' point is given in terms of th e peak value ofsupply current as expressed in mao

For B2 max =20 volts, I =20/0.9 = 22 mao Considering a.11 thecurrent as magnetizing current, an approximation for the dBmaxi s giTen by.

All = J.L 6H = 0 . 4 T T N I ~ max e 1 emt r

0.4TT(24) (22) (450) 10-3 == - - - - - - - 133 gauss,.2.24so that the core is well below saturation.

Next the 50 microhenry tail-reversing inductance Lt was consideredto be in parallel with L,a s shown in the top of Figu.re laB with switch Sclosed. Mea.surements showed that this inductance a distributed capacitanceof 4 mmf, so this was combined with C2 to give the following 'new values forthe 1:1 equivalent circuit:

found:

Cl = 23 mmf.02 = 57 + 4 =61 mmfJL = 15 I)licrohenries_ SO x 95 _L - SO + 95 - 32.8 microhenries

15 - 4(j= 15 + 32.8 - ,0.31From these new parameters the natural frequencies of resonance a.re

7wl = 5.4 x 10w = 2.24 x 10720( = 2.25 x 1015

= 1.45 x 1030~ ' . =6.45 x 107llJi' = 1.85 x 107x' = 2.05y = 0.59K = 1.54 x 103


40/87

Report R .217N ~ t e that th e addition of th e shunt inductance Lt has l i t t le effect

upon the high frequency W;a" , but increases W;a'. The dashed curve of Figure 18shows the calculated response. Here again the plot is given on the same scaleas the solid curve 80 that K for the diagram is 103

F , i ~ r e 188 may be compared with the observed performance in Figure 26for Ferramic H, and i t will be seen that there is good correlation betweenthe calculated response and the observed response. The effect of the shuntinductance Lt in i n c r e a s i n g ~ ' has caused the "ringingA frequency WR I I to becontained below the voltage base line after the ini t ial positive pul.se. Theeffect of the damping diode is then to overdamp WR I sufficiently to preventthe recurrence of a positive pulse.


41/87

Report R-217CHAPTlDll IV

PULSE TBANSFORMmR CONSTRUCTIOH AND !.msr

A. CONSTRUCTION OJ' PULSE TRABSFOlUmlSThere are three general geometric shapes which are available. for

pulse transformer construction. These are the C-shape, the cu.p core, andthe ring or toroidal shape. The C-shape is advantageous when an air gap isdesired and has the further advantage of permi tting the use of coil formsfor the windings. The disadvantage of the C-shape is that i t requires a c ~ i p of some kind to hold the C-shapes together.

The cup core is being used extensively for small pulse transformersby several manufactnrers because of the relative ease in assembly as comparedwith the C-shape. The typical procedure is to provide a hole through th ecenter of the cup shapes so that a mounting screw may be inserted fo.r c1ampingthe two halves of the cup s h a J ? ~ together. The disadvantage of the cup coreis i ts relative high cost. For WWI the core had the further disadvantagethat the leakage inductance could not be controlled as easily as the C-shapeor toroid. A fnrther d i s a d v a n t ~ e is the necessity of bringing the coilleads out of the cup without lead-to-core breakdown

.For WWI the toroid had the advantage of small size, low cost, andease of control of leakage inductance. The difficulty in winding procedarewas in a large measure reduced by the concurrent development in the Labora-tory of a toroidal winder which could handle the small 3/16" inside diameterr ing. For these reasons the toroidal shape was selected for the 3:1 andthe 1:1 ferrite-core pu.lse transformers.

This winder and the assembly technique for the pu.lse transformers weredeveloped by Mr. Robert Hu.nt of the Digital Computer Laboratory, .M.I.T.-35-


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Report R-217 -36-

F i ~ e 19 shows the assembly of a 3:1 pulse transformer. The toroidal shaped core was f irs t put through a tumbling mill to knock down thesharp corners and then the core was coated with enamel to prevent chafingof the wire during winding. The primary and secondary windings were thenplaced the core. For the 3:1 transformers the windings were concentrically wound, which for the 1:1 transformers the coils were p1a.ced on oppositequadrants of the ring. The wou.nd transformer was then placed in a phenolicmounting board which contained the heavy pigtails for external connection.After soldering the transformer wires to the pigtails the whole assemblywas placed in a die and a thermo-setting plastic resin used to mold a pro-tective cover over the tr.ansformer. The complete transformer which m e a ~ r e s 3/4" x 3/4" x 3/g" resembles a condenser with four pigtails.:So TEST CmCUITS FOR PULSE !f!BANSFOBMERS

In the preceding chapter on design mention was made of the testcircnits which were used as the basis for determining the performance of the3:1 and the 1:1 pulse transformers. These circnits were selected as beinga typical application for the transformers in WWI and were constructed intoa test panel for rapid transformer testing. The c i r ~ i t s are shown in Fig.20 and will be discussed now in order of their appearance from top to bottom.

The 1:1 pulse transformer circuit is a basic gate tube circuit . Innormal operation a gating pulse will be applied to the suppressor which permits the input pulse to appear across the tube output. .For the test circuitthis suppressor was grounded to the cathode. Two switches were provided topermit the observation of output waveshape with and without the damping diodeand with and without the ta i l reversing inductance.

The 3:1 pulse transformer circuit is a basic point-to-point amplifier circuit using a power pentode with a 250-volt plate supply. In this


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Report R-217 -31-

circuit a damping diode was ~ o n n e c t e d through a switch across the primar,y, , ' ,

of the transformer to demonstrate the increased sensitivity to pulse r e p e ~ t i ion frequency when the overshoot is clipped. Not shown in the 9-iagramis a 93-ohm resistance load across the transformer output.

Also included on the test panel was a gate generator output amplifiercircuit which is used for checking 5:1 pulse transformers. (This thesis doesnot include a design for 5:1 transformers. In application these transformersare driven by square pulses in the order of 0.5 microsecond duration.)

Figure 21 is a block diagram of the eqUipment used for supplyingthe input pulses to the test circuits. A variable-frequency clock-pulsegenerator provides a constant series of half-sine pulses of 0.1 microsecondduration. For most of the test work the pulse repetition frequency was setat 2 megacycles. 'The output of the clock-pulse generator was fed to a scopesynchronizer where a frequency division occurs to produce positive pulsesfor triggering the synchroscope. The internal trigger delay of the synchro-scope was used to provide an adjustable delayed pulse to a gating pulsemultivibrator of a pulse ~ t e r . The gate tube of the pulse gater receivedthe gate pulse on i ts suppressor and a clock-pulse on i ts control grid. Theresultant chain of O.l-microsecond half-sine pulses were then fed through thepulse amplifier of the pulse ~ t e r and then to the pulse transformer testpanel.

F i ~ r e 22 shows the complete test set-up, excluding the synchroscope.The transformer connections were brought out to four-terminal Jones plugswhich were mounted in the front of the panel. Thus the experimental corescould be quickly tested using a standard plug-in j ig .c. TEST RESULTS

Figare 23 shows a comparison of various ferrite materials used forthe core of a 3:1 pulse transformer. These materials were wound with 21 and


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Report R-217 -38-

7 turns and the o u t p u t ~ a v e s h a p e s are shown under various load conditionswith the IVI BYPersil-core transformer as a reference. Considering the opencircui condition f i rs t , the leakage inductance in the WI transformer is,seen to produce a small ripple or oscillation on the waveform which is absentin the other transformers. Also, the eddy-current losses in the WWI B1,persiltransformer produce a greater damping effect on the low f r e ~ e n c 7 oscillationthan that which occurs in the ferrite transformers. The effect of a decreasedpulse permeability i s shown in the comparison of the Ferramic H, Ferricore,and MFl13l. These materials ~ v e pulse permeabilities of 450, 230, and 40r e ~ e c t i v e l 7 , and the resnltarit decrease in positive pulse amplitude and increase in overshoot amplitude are noticeable. MFil3l is the square loopmaterial with the long time lag mentioned in Ohapter 2, and i t is significant that the low frequency oscillation is more effectively damped thanFerricore.

When a 93-ohm resistive load is connected across the terminals ofthe transformer the normal operating conditions are simulated. The FerramicH transformer produces the type of o v e r s ~ o o t which was desired in order togive recovery before the next pulse. The Ferricore and MFll3l transformershave faster recover,y times but at the expense of positive pulse amplitudereduction.

I f a damping diode is connected across the transformer, the over- 'shoot is "clipped". While this appears to be an advantage i t is only necessary to recall that the recovery time constant is L/R, where R is the totalresistance across the transformer during overshoot, to realize that ~ c h adiode will produce increased sensitivity to pulse repetition frequency. Thisis brought out in Fig. 25 where the effect of the damping diode on a chainof pulses is observed. With the diode in the circuit there is a decay inpositive pUlse amplitude which is absent when the diode is absent. This


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Report R-217 -39-effect is even more noticeable in Fig. 24 where the integrated output isshown for the cases with and without the damping diode. With S closed, thediode is in the circu.i t and the integral. of v t is seen to approach zero mu.cb.slower than with S open.

F i ~ r e 26 shows the output waveshapes of the 1:1 pulse transformermade of Ferramic H as compared to the WWI standard which has a Hypers11 core.The Ferramic H waveshapes are in close agreement with the analysis of OhapterIII , and the waveform is seen to be a good approximation to the WWI standardta i l reversing inductance and the damping diode are in the circuit . In the.case of the 1:1 transformers, the recover.r time with the diode in is snfficiently fast so that the pulse repetition frequency sensitivity is not ofconcern.

From Figs. 23 to 26 i t can be seen that pulse transformers made withferrite cores can be designed to replace the Hypersil ones. Although thecores used in the design were Ferramic H, other materials such as Ferroxcu.be30, Oeramag 51' and Ceramag 7A. will perform equ.a1ly as well. The final selection of material from this representative group may then be based primarilyon cost.

In the 1:1 transformers, i t is possible to "build" the ta i l reversinginductance into the transformer by using a lower permeability core materialthan Ferramic H. Experiments made with Ferricore indicate that this materialhas about the right value of pulse permeability to provide the decrease inself-inductance that is required. (:Because the leakage inductance dependsupon geometry, using the same toroidal shape and winding scheme thus permitsthe leakage inductance to remain constant.)


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Report R-217CBAPl'ER VSUMMARY

The main objective set forth for this thesis research was to determini f the new ferr ites could be used as core materials for pulse transformers forWWI. The research consisted of (a) determining the magnetic behavior of theferr ites under pulse conditions, (b) developing an equivalent circuit whichcould be used to predict the response of th e pulse transformer under WI condit ions. and (c) designing and testing ferrite-core pulse transformers. Themethod of attack and the results obtained from each of these three parts ofthe research will now be summarized.A.DETERMINING THE MAGNETIC BEHAVIOR OF FERRITES UNDER PULSE CONDITIONS

Apparatus was constructed to perform pulse tests on the ferr i tematerials. This equipment consisted of a gas-tube pulse generator whichprovided a variable pulse width of from 0.1 to 0.3 microseconds of half-sinecurrent for the core excitation. Using a standard procedure of integrationand amplification through video amplifiers, th e pulse loops were presentedupon a cathode-ray tube and. quantitative measurements of A B V S. L1 H taken.When these data were compared with data taken from slow-speed pulse excitat ions, i t was found that there was good correlation. This indicates thatat the 0.1 microsecond pulse widths th e eddy current effects are small and th eferr i te material can be judged on the basis of i ts d-c characteristics. Valuesof pulse permeabilities were then given which can be used in pulse transformerdesign. I t was shown that for pulse widths less than about 0.5 microsecondsthe ferr ite materials have higher lnlse permeabilities than grain-orientedsteels.

The disadvantages of the fer r i tes for pulse transformer applicationswas pointed ou t to be (a) low saturation values, and (b) low Curie temperatures

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Report R-217 -41-

For the application in WWI, however, these disadvantages are not particularlyimportant.B. DEVELOPMENT OF RJLSE TRANSFOBM1S1R BQUIVALICNT CIRCUITS

Because the two transformer designs presented in this thesisinvolved entirely different operating conditions. two separate equivalentcircuits were developed. For the 3:1 pulse transformer which operates intoa low-resistance load. th e leakage inductance was made small order toprevent attenuation of the output voltage. This permitted the circuit representation by a simple LRC parallel combination for the 3:1 transformers.

For the 1:1 pulse transformer which operates essentially opencircuited. use is made of the natural frequencies of oscillation of th etransformer to produce a large positive pulse. For this reason, the leakageinductance is purposely made f ini te and the equivalent circuit for the 1:1pulse transformers, which consists of a series R 1 in combination with aparallel LC 2 , i n c l u d ~ s this element.

Before the actual equivalent circuits were developed a "BasicTransformer" was introduced. This consisted of a pure lossless inductanceand i t was used to i l lust ra te the relation between the voltage-time planeand the B-H plane. Kffects of non-linearity in inductance were shown for theconditions of no-load and full-load on the Basic Transformer.

Following the introduction of the equivalent circui t for the 3:1pulse transformer, analytically derived curves were presented which could beused in the design procedure. These curves show the effects of resistiveloading upon output voltage wave shape and magnetizing current for the conditions of a half-sine excitation. The effect of capacitance upon output waveshape was included and a hypothetical ease of non-linearity in inductance wassimilarly shown.

In WWl.advantage is taken of resonant frequency peaking in the 1:1


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Report &..217 -42-pulse transformers when operated under no-load conditions. For this reason,the 1:1 pulse transformer equivalent circuit was developed with the objectivebeing to disclose the nature of the ,natural frequencies of oscillation in thesetransformers. Two methods were then presented for determining these oscillation frequencies. One consisted of a graphical method of susceptance plotswhile the other was an analytical method. Formulae were then developed fordetermining the output wave shape of the 1:1 pulse transformer under no-loadconditions when driven from a half-sine current source. From these formulae,curves were plotted of th e output voltage wave shape for various ratios ofthe resonant frequencies of the transformer to th e fundamental frequency ofthe driving source. These plots serve as a guide to indicate th e properdirection in a design.C. ;DESIGN AND tESTING OFF.ERRIT.Ii1-CORE PULSE TRANSFOR)/lERS FOR W I

Using the equivalent circuits, a design procedure for a., 3:1 pulsetransformer to replace the Hypersil-core tra.nsformer in WWI was presented.This design was based on a low pulse r e p e t i t i o ~ frequency sensitivity criterionThe response of t h i ~ design was then compared to th e Hypersil c ore and shown tobe compatible. Effects on pulse repetition frequency sensitivity of a dampingdiode were also shown.

In the 1: 1 pulse transformer design, the criterion \m S to duplicatethe voltage wave shape of the WWI Hypersil model when a damping diode and t a i lreversing inductance were placed across the transformer. In order to do this,i t was necessar,y to determine experimentally the required number of turns.From the computed values of self-inductance and measured values of l e a k a ~ inductance and circuit capacitance, the wave shape of the ferrite-core transformer was then determined analytically and shown to closely resemble th eobserved actual response. The equivalent circuit was thus shown to be adequatefor this application and will be valuable in predicting th e performance of


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Report R-217 -43-the 1:1 transformer under circuit conditions other than that for which thetransformer was tested. The 1:1 equivalent circuit is further helpful inbringing out the fact that to eliminate the ta i l reversing inductance oneneeds to keep the leakage inductance th e same while reducing the self-induct-ance of th e transformer. Since both inductances are proportional to thesquare of the turns, the way to do this i s to use a ferr i te with a low valueof pulse permeability.

The method of fabricating the ferrite-core transformer was shownwhere a thermo-setting plastic resin i s used to cast a protective coatingaround the unit. The resultant transformer then becomes small in size, l ightin wei&ht and easily ~ o u n t a b l e . No figure was given for the comparitivecosts of the ferrite-core transformers and the H ~ r s i l - c o r e transformersbecause complete figures were not yet available at th e writing of this thesis.I t is expected, however, that the ferrite-core transformer will cost only afraction of that of the metallic-core pulse transformer.

In conclusion, i t can be stated that the ferrite-core pulse trans-former appears to be admirably sui ted for 0.1 microsecond. pulse applicationsin WWI. There are at least four and possibly more ferr i te materials whichare suitable for this work and th e question of selection becomes simply aquestion of price. With regard to the shape of core, the designs presentedhere were based on a ring or toroidal shape However, having determined thenature of the problem and found the resultant equivalent circuit and p a r a m ~ t e r values necessary to produce the desired results , there is no reason to believethat other shapes such as C-type and cup-core type could not be used. Inthis research, some difficulty was found in controlling th e leakage inductancefrom the negigible value required in the 3:1 design to the comparitively largevalue needed in the 1:1 design when a cup core was used. (For purposes ofstandardization, the shapes for the 1:1 and th e 3:1 pulse transformers should


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Report R-217 -44-be the same.) In the C-shape. however. the primary and secondary windingsmay be placed on opposite legs to introduce the necessary leakage inductancefor the 1:1 design. Specifications for a C-shape core were drawn up at thebeginning of this thesis work, but unfortunately samples had not yet arrivedat the writing of this report.

ApprovedJay W.


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Report R-217APP.iNDIX A

APPARATUS FOR TAKING PULSE HYdST.lBESIS .,LooPS

In order to generate half-sine pulses of variable amplitude an d,,..,pu];se width for supplying the H excitations to fe'rrite c o ~ s , th e circuit

of Figure 1 was constructed. In the plate circuit of the 2D2l hydrogenthyratron, the switch S selects an inductance which together with capacitoro determines the time duration of the generated pulse. A 100,000 ohm potenti-ometer in the plate supply determines the pulse amplitude, while the 200,000ohm series resistor is necessary for isolating the plate circuit from thesupply during the period of tube conduction.

To drive the generator, a pulse from a P-5 Synchroscope ~ a s usedat a repetition frequency of 4,000 cycles per second. This pulse was dif-ferentiated in th e grid circuit to give a sharper peak, the differentiatingcircuit consisting of the 68 mmf condenser and 10,000 ohm resistor. Anadditional 10,000 ohm resistor was placed in series with the grid to preventexcessive grid current flow.

The sample was placed in the cathode circuit of the pulse generatoralong,with resistor Rl for measuring H and the integrating circuit R2 and O2for providing a voltage proportional to B. The value of Rl was experimentallyadjusted to cause i ts voltage drop to be approximately or less of the totalvoltage across the sample. For this reason, the integrator can be consideredto, be effectively across the sample alone. The value of th e integrator elementwas determined by placing a minimum load condition of 10,000 ohms across thesample winding and to give an integrating time constant of J.J microseconds.For 0.1 microsecond pulses which have fundamental frequency component of5 megacycles t the integrating wRC = 16.5 i s well above th e wRC =- 10 which i snecessary for better than 1/2% accuracy. Although i t would have been desirable

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Report R-217 -46-

to i ~ c r e a s e R2 to 100.000 ohms, this would have necessitated C2 being JJ mmf.a value comparible with stray c'apacitances.

The voltages from C2 , representingB, and from Rl , representingH, were then amplified through two video amplifiers with identical time delays.For this purpose, two Tektronix 5l4D scopes were matched, the amplifier ofone being used to drive the horizontal plates of the other. Since for th esize of the sample and th e number of turns used, the values pf B and Hvoltages were in the same range of magnitude, this proved to be a workablemethod. The scopes were oriented with respect to each other in such a manneras to permit the use of short intra-connection leads. Although the manufacturespecifies a bandwidth of 10 mc. for these scope amplifiers, this is not anunreasonable value since the H-wave shape is half-sine with a fundamental

frequency of 5 mc. The B-voltage bandwidth requirements are of course notas stringent since this is an integral. For reasons of amplifier distortioni t is to be expected that the pulse hysteresis loops will not show theirtrue sharp corners.

The calibration of this equipment was extremely simple since th eoscilloscopes were provided with voltage calibration controls. The followingformulas were used to ca.lculate B and H:.

_ N v lH - 10 lmRI oersteds8B = 10 R2C2v2 gaussNA

1 =mean magnetic path (cm)=number of turnsvl = voltage across RI (volts)v2 = voltage across C2 (volts) 2A = cross-section area of sample (emR2C2 = integration time constant (sec)

Glasoe and Lebacqz, Pulse Generators, Rad. Lab. Series No.5. p.642. McGraw-Hill Book Company. 1948.


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Report R-217APPENDIX B

3:1 PULSE TRANSFORMER EQUATIONS" .: ,. ": :.',

1. DERIVATION OF EQUATIONS FOR PLOTTING FIGURE 13

xi IRFig. B-1 Fig. B-2

Figure B-2 i s equivalent to Figure B-1. In both f igu res ,a l lvalues are referred to the secondary of the transformer. W ~ i t i n g .theloop equations for Fig. B-2 in Laplace Transforms:

L s I ( S ~ (Ls+R = Et,(s)Since I( s) = IRW (1+ e tf S) (the transform of a half-sines2+W2 '

pulse),Et,(s) =

and!ti.)Ls

From Formula 1.210, Gardner and Barnes*;For 0 S c.ut 6 rre(t) =Er(t) =

(IJ -1 Rwhere! = tan -. t -1 R- an -]{---------_ ...-----.......- - - - - - - ...---------_ .._--------

if- Gardner and Barnes: Transients in Linear Systems, John Wiley & Sons, 1942.. -47-


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Report R-217 -48-and

From Real Translation Theorem;

Therefore for 1T wt, de fine l4)t' wt - Tr, and?.. . Rwt'( ) ( ) ( ) IR-u.a.. - ""WL f,e t = EJ. t +E2 t = - R2 + (WI.)2 e t t e "oW (la)

im = -e(t)/R (2a)2. DERIVATION OF EQUATIONS FOR PLOTTING FIGURE 14

Figure B-3In figure B-3 all values are referred to the secondary. From

the node equation in Laplace Transforms;E(s) =

The roots of the characteristic equation for the overdamped case are:1 - ~ J C J 2 12RC - -O1 + V ( ~ C ) 2 - 12 2RC LC


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Report R-217For 0 fAt ~ 1 T

~ ( s ) = !o

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','

Solving for the residues, the terminal voitage can be expressed as:

where

l\[ j fA} ej wtl _c2J. (oc. t jw) ( ~ ~ j(j)) J -Since ~ ( s ) = :&t(s) ,

For 11'4wt , define cut' :w t -1T Using th e Real Translation Theoremand combining terms:

et =El(t) t E2(t)_ lK1 a-lICit' (1 t a-dol/!,) t K2 a-a(Lt '(1 + e-o(l.!j)] (3a)

i _ l - ~ e - ~ l t ' , ( 1 + e . . . o l , ~ ) - e-o(lt f (1 t e - ~ 1 (4a)m - 10


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Report R-217 -,50-

3. DERIVATION OF EQUATIONS FOR PLOTTING FIGURE 15

Figure 13-4The conditions for Figure 14 for R = 3 ohms, e - 90 mmt, and

L = .4 microhenries are asswned to exist for 0


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Report R-217 -51-4. DEVELOPMENT OF DESIGN FORMULAE FOR 3:1 PULSE TRANSFORMERS

Inductance L is defined as flux linkages per ampere:*L = .tlli xlO-S - xlO-8 henries

im - imwhere

R-217 Design of Low-Power Pulse Transformers Using Ferrite Cores Nov52

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