Design of a self-saturating magnetic amplifier utilizing high frequency excitation
Item Type text; Thesis-Reproduction (electronic)
Authors Perkins, Brayton Mills, 1924-
Publisher The University of Arizona.
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Link to Item http://hdl.handle.net/10150/319332 DESIGN OF A SELF-SATURATING MAGNETIC AMPLIFIER UTILIZING HIGH FREQUENCY EXCITATION
by
Brayton M. Perkins
A Thesis
submitted to the faculty of the
Department of Electrical Engineering in partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE
in the Graduate College, University of Arizona
1956
Approved: C a t * * \ . / f f Z Director of Thesis Date
5 ^
This thesis has been submitted in partial fulfillment of requirements for an advanced degree at the University of
Arizona and is deposited in the Library to be made available to borrowers under rules of the Library• Brief quotations from this thesis are allowable without special permission, provided that accurate acknowledgment of source is made.
Requests for permission for extended quotation from or re production of this manuscript in whole or in part may be granted by the head of the major department or the dean of the Graduate College when in their judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author. SIGIpD: D1SIG-H OF A BH»F=>S&TOMflHa MM&mTXC 1SPL1FIER UTILIZING- H G H FREQUENCY EEGITATION .
Drayton Perkiias
transfer .fiiBotion for a self-saturating (with low eontrol eireuit resistanee) is
.9 the elementary parallel-eonneeted saturable
.reaetor amplifier as a basis0 Other equations pertaining to
operation and.'design ©f a self~saturating amplifier3 with a
high frequeney power supply ( 1 0 ko. to 5 0 ke0) g. rare d.©riTed„ A simplification of the tlm© eonstant arid the gain expression
in the; transfer equation is earried out0 A sample design pro- .©edure'ls set up for the design of an amplifier by running relatively simple tests on proposed, core material and diodes and by using the equations developed; then an aetual design is earried out step: by step to.illustrate.tM sample procedure An experimental bridge self-saturating magmetie amplifier: was
©onstrueted from the design; and a eomparison of D 0G 0 input- output ©haraeteristies and frequeney response ©urves were made ''between' experimental results and these predieted by equations = Other ©omparisoris' are made 'eoneerning magnetie feedback and
©hange of input resistanee0 All results appear to eheek with in the limits of the approximations mad© in the equations.o AGKHOELEDG-Siirf.
The author wishes to thank Professor 0o Eo Bunn for supervising the research and the preparation of this'thesis= The research was condnoted using the facilities of the Electrical Engineering Department at the University of Arizonao /v , , ' ' ; 0O1TEEES „ '
I M E H J O E E O H , Pag©; ,. Chapter lc THS SATOEABLS RMGTOR AS A COKTEOL DEVICE 1 .
1 o 1 « iBtretoetiom' . : \ , 1
l 0 2o" Diseussion of Core Materials . 2 lo3o laduetanoe as the Amplifier Controller 4 lo4o Method of Varying Induetanee 7 Chapter 2, Sm'-SAIUBikTIlMG MGKBTIC AIvlPLIPIlR TMNSFER . rnwgmow 1 1 2^1P Introduotion 11
2 o 2 o The Par all ei«- connect ed SR Magnetic. Amplifier 12
■2*3*. : The General Transfer Function 21 2o4o Modification of the Transfer Function Time Constant 2S 2o 5o Ideal lnpu.t=output Characteristic for the Self^saturating Magnetic:Amplifier '32
2 0 6o The Modified Input-output Characteristic 4 0 2o7o Further Determination ©f the Transfer Funotion Variables 4 4
Chapter 3 = D E V m , O P m m OF DESIGN EQ.UATIONS AM) A DES1GM PR0C1DUR1 47 3ol» Introduction 47 3o2o Simplified Reference Gain 4$ 3<.3= Application of the Simplified Reference Gain function 53
3*4o Simplified. Time Constant 57 v ./ v.; ; ; , '3 = 5 =. SB Ixitation Reqmlremettts ■ - - 58 3 = 6 o A Sample Design Procedure 66 eripter: 4 EEPpiiMEZPii, aisJBfS . . ' . 77 4*1 = Intx-saudtidn . . - 77 • 4=%= ' Ixperimentai Test B@suits: on 6 @re Material 77
' 4 o 3'6' Bzperimental $est;Bestilts on DleS.es . ■ 84 "4 5=2= Other Remarks : 7 ; "' ; ' , - 1 1 8 ■ iPPiEDix 4 : ; ' 1 2 1 • ;'BlBH0dB4EW . :;y ■ ; ;v::: ; . v ' : : :M2 8 i d Ideal Hysteresis' Loop . - ; . : 3 . lo2 Non-ideal Hysteresis Loop : 3 1*3 Elementary Magnetic Amplifier J '5 16,4 Hermal Magnetization Oia,rir@ 4' : 5 1»5 Flux Density Components Present in Ferro- aagnetie Materials , ■ - ' .. g 2 01 Farallel-00nneeted SB Magnetic Amplifier ; . 13 202 Belationskip Between Core Flux ( }■ and SB . ■ . ‘ Toltag© i v ^ ; >T ; ' T 1 3 2 o 3 Average Value of SB Teltage ‘ 18 2Q4. - Eodp Ourrerits in a Parallel-eonneoted SB . ■: Amplifier . : i .i / ' 18, Zo 5 Sign convention for Amplifier Feedback . 22 2 C 6 S e r w Bloek Diagram of Farallel^eomeefed. ' SB Magnetic Amplifier ' ' : ; ■ 22 2o7 Servo Bloek Diagram of Self-saturating Magnetic , : Amplifier •; . . . ; . ; 2? 2oS Simplified Servo Bloek Diagram of Self- A ■ . saturating Magnetie Amplifier ' - 27- 2o9 Magnetic Amplifier Transfer Function Table 2 9 Sold Tke Equivalent Input Circuit of a Magnet!© ; -I" Amplifier ' - l i’: - 3 0 2 o 11 % e 'Bridge Self-saturating Magnetic Amplif ier y- for High"Frequency Operation 3 0 2ol2 Waveforms for the Bridge Self-saturating . Magnetic Amplifier ' 3 3 2 oX3 Magnetizing Gurrent Supplying Ezoitation .. _- -to - ^ v V "3;^:.: : ■" 35- : 2014 Magnetising Current Supplied, by SR& to ...... : 35 2oX5: : Saturation Load Current Path During a Half - ::: CyeXe of: Ezeitatio'n ■ / 39 2®X6 Beetifisr Heferse Current for Half=eyele of ' 42 : 'Sol ^ TypieaX D 0C 0 Input-output Char act eristic :: • 9 . : . Curve ef Self ^saturating Magnetic i,mpl i f i er 49 3o2 Straight Line Approximation of D 0C a Input- output Chara'etefistio with.no Feedback Present . 49 3o3- Straight Line Approximation of Input-output ,: . ' :: Characteristic (Showing Construction Lines for • ■ Graphical Determination of Characteristic with ■ . , Magnetie Feedback Only) • ", " .. . 54 3o4 Straight Line Approximation of Input-output Characteristic (Showing Construction Lines.for Graphical Determination of Characteristic with , ■ : Rectifier Reverse Current Present.) : 5 6 3o5 Magnetizing Current of Self-saturating Magnetic 3o6. IxampXe'of 6 Sd and Even Punetions .' 6 0 3o7 Magnetizing; Current and its Fourier Approximation6 3 3 o-8 . Total Path of Magnetizing Current for a Half1 • Cycle of Power Supply Operation for the Bridge • Self-saturating Amplifier (Only Magnetizing : . 1 1 Current Flows. ' .t ,v- 65 " 3»9 . Flux Hecessary for Mormal Excitation of SB 6 5 3 10. Choi 6 ©, of Design DoG c Input-output Charact eristlc 6 8 : 3oil Example:of possible Core and Sinding Cross FIGURE " . lllnstratlom of Fe©dba.ek la Bridge Self= 3 o5-2 — Jt uL- t) w it> u * ‘ft ^ e* « ft ' 75 : " 4ol ; let hod of Measuring the jGop of Gore Material 79 4 o2 ■ Gore Magnetizing Current and Output 79 sis Loops for 2 Mil Super mall oy Tape 8 2 4°4 - Differential Permeability obtained from Parallelogram Approximation of Hysteresis 4»5 ;e Test oil Diodes at a Frequeney of 30 K e 0 83 4o 6 yariation of Differential Permeability with Fregneney for a Core Speeimen of 2 Mil Tape _ ' -- : S5 Hysteresis Loops (Parallelogram 1 for Sup email ©y at Two 8 6 408 Dynamie Hysteresis Loops Approximation) for Supermalloy at an ; Ff eq.u©ney of 30 K © 0 . -. 87 409 Mot hod of Determining Diode Equivalent 4ol0 Method of determining Average Value of A;'- Eeotifier Reverse Current ' 88 4 dll. .;F 6 f 8 of Three Different Diode 90 4*12 Photographs Showing Eeotifier Reverse Current 91 4ol3 ; Representative Plot'of Sinusoidal Load Current - : • Versus Eeotifier Reverse Current 4*14■ Reetifier Reverse Current for- Two Different ■ Diode. Types - - - 94 4sl5 Straight Line Approximation of Amplifier D 00 o Input”output Cliaraeteristi© FiGmih ' . : , ; :. pag@ ::- 4®16 'Amplifier Do Co Input=output Oharaeteristie '(& Bi©des>,. ' - : 1G2 46lf Gomparlsdii of i)eslgn Ireqmenef' HespoBse with Mzperimemtal Freqwiiey Reapeaae 103 . 4olB 4tepllfler Do Oo Xnput =• output Charaoterlstlo IB :Di©a©s) . - , 104; , ,, 4el9 Straight Line Approximation of Amplifier IX Co Input-”output. GharaeteristiQ (B Diodes) 106 4o20 Ampliiier i)0 Co Input-output Charaoteristie :, (C Diodes} ' , ' - ■ . 10? 4®21 Straight Line Approximation of Amplifier Do 0 0 . Xnput-dutput Charaeteristio { 0 Diodes) 108 4o22 A New Approach to Supermalloy Hysteresis Loop ■ r. ■ - - . Approximation - - - ' V ' ' 109 : 4®23 Photographs of Amplifier Output 111 4a24 Straight Line Approximation of Amplifier D* CB 1- , * ■. Input ^Output; Characteristic. (Infinite Gain) ■ ' -112 4o^$ Straight Dine Approximation of jWplifier Do: Ca ' . Input^output Characteri'sti.o.. (Finite Cain}, ■ 114 48.26 Effect of Input :Resistance on Frequency; : ;/: Bespohse of Self^saturating Kagneti© Amplifier - - ’ for a Saturation Angle i 1 of 127° • 1 1 5 - 4®2? Filter Used on Output of Experimental Amplifier 116 5*1 Experimental Amplifier DoCo Input-output Characteristic Showing How Assumption of no v : .Diode Reverse Current Results in Error» ; 120 Appendix 'ol A Toroid with IP Layers of Wire Wound in a I v Circular Manner - - . ' v ( ...' ■ 122 Appendix; e'2 A Toroid with a Single Layer of Wire Wound ~ . .to Conform to the Gore Configuration ; . _ 122 INTRODUCTION ' One of the eMef advantages of using magnetie amplifiers ha s. been ruggedness and reliability 0 Power sources In the low ■ frequency range (60=400 cycles per second| have been no prob-- lam3 but when requirements demand higher frequencies9 greater difficulty is experienced*. Until recently there were' few power supplies capable of operating at high frequencies with equivalent reliability and ruggedness« ; ■ Several papers are now indieating means of using fre quency multiplier techniqueso • Oo -H* Royer ; describes a . . ; switchii^ transistor-D* Op to Ao 0 * converter which has an - output frequency which is proportional t© a Do 0 * input voltageand 'this;devie© appears to have promise in the' : future* ^ •.: ; ; . Available literature appears to Indicate a lack of informa tion concerning the design and. operation of many types of mag netic amplifiers at higher excitation frequencies« ; In view of this fact it is proposed to choose a standard • type3. the self-saturating magnetic amplifier with; low control ,circuit resistances, and "investigate its operation at higher . excitation frequencies f 1 0 = 5 0 -kc.} as a voltage: gain-, device. for low.input signals * ' Go Ho Switching Transistor D-G to A-G Oonverter an oy Proportional to the D-G Input Voltage^ A1 ES Transactions 74s pt,Is duly3 1955» pp* 322=26<, C H M S E 1 ,/ p ® m m m B m m m T m m bihgi. . ilo1o) ■' IntroduGtion There are at least two main classes of magnetla 3'. ■ ■ - . • * . ' amplifiers, s. .. ■ ‘ . Ca:») Saturable reaetor amplifiers eontrolled by a signal magnetomotive foroe (a 0 e 6 excitation is applied t© the core by the load or reactance winding)„ (bo) Saturable transformer amplifiers controlled by a signal magnetomotive fore© |"a»©« ©xioitatlon is applied to the core by a separate: primary winding)® 'r ; ?: It is the purpose here to consider only the class listed as ,, saturable reactor'so Falling in the group ©f magnetic amplifiers under consideration^ are such basic types as the series connected9 the parallel connected9 and the so-called self=saturating amplifiers a all of . which ©an be operated with free, even-harmoni© or with suppressed even-harmonic currents in the control circuits® Free even-harmonic current operation implies low control circuit resistance3 and suppressed even- harmonic :current operation implies high ©entrol circuit re- : r-v'/-- • : - ' ^'T 'I - sistanee o Indications of suppression are usually given by a 1 ' . 1 0 Jo G-o Miles 9 89Types of Magnetic Amplifiers,w h U E Trans® <, Vol® 71» ptb I s 1952$ ppc 229-38» : : , / ■ . • . ... a suppression inti ex* - - Of t,he types mentioned, above, ■ investigation, of high i frequenor ataptation will, be •essentially limit ©a to a study ■- of the self^saturating amplifier using low control circuit • r©sistanceo-. \ ' " \ C1 o2 <,) Discussion of Gore Materials ' . The basis element,comprising the magnetic amplifier is, of sourse§ the Saturable reactor; and the heart of the satur able reactor is Its core* : " A study of the saturable reactor core willg of necessity, . concern itself with a study of the core hysteresis loop0 The most desirable hysteresis; loop that could possibly be obtained. would-hays the form shown in Figure 10^ Figurel=2 more accurately-ahows the hysteresis loop as it is found in prac- - tice<, -Bo matter how closely the loop approaches the ideal, as soon as the frequency is increased, widening of the loop ' OGCUrSc . ; " • ; .' Methods to reduce eddy current losses to assure faithful . transmission of information and t© reduce heating has led'to such core configurations as magnetic powdered cores {impracti cal because of three dimensional subdivisions of the material) and thin punched laminated cores Cimpractical from the stand point of fabrication, handling, and insulation)» laminated ceres have the additional drawback that ag laminations are. . . i. Ho Ho lord, ^Dynamic. Hysteresis Loops of Several; Gore Materials Employed in Magnetic Amplifiers**, AI11 Transactions, ' Vols 72, ptc 1, 1953, PPo 85-8S0 3 — *S<>/css'<=r T/as? /-/z/jc J7d>r7j-//y Bs //7 (Sarcsjj H // - /^T^/7 £*/<3 S^OZ-Cf? ZZ7 /<:* / y a a / ' t p // / / % % //yj-rzy?£ir/j z w r /v&c/r The present trend appears to be toward permalloy cores of tape (2 mils and less in thickness), With this type of core a median is reached between the need for small eddy current losses and the need for a magnetic circuit as nearly closed as possible. At present, high quality cores of thin magnetic tape are being manufactured commercially using cataphoresis (deposition of charged colloidal particles under the influence of an electric field) as an insulating method.'* At any rate, it appears that the hysteresis loop characteristics determine to a great degree the characteris tics of the magnetic amplifier. (1.3.) Inductance as the Amplifier Controller FigureU depicts an ideal sinusoidal voltage source supplying excitation to a saturable reactor and a load re sistor in series. (Hereafter, the term saturable reactor will be designated only as SR.) The equation of this circuit is ex = vtRL + L ^ - yt (Rl * pL) 1.1 p is an operator, ^ 4*A. G. Ganz, "Application of Thin Permalloy Tape in Wide-band Telephone and Pulse Transformers", Electrical Engineering. Vol. 65, 1946, p. 177• 5*H. L. B. Gould, "Magnetic Cores of Thin Tape Insulated by Cataphoresis", Electrical Engineering. Vol. 69, 1950, pp. 544-548. 5 J’/r7 p ^ J *-- >/'?L 1 /.3 B o L?/ (T /rl j/c’S~&J/ S / oeyy /y<3/-/77&/ /*/**?/*t?'7C/2: /*/&/? Z^/ytsre A 4- 6 Solving for i and multiplying by gives e* ■ ex rL Rl + Lp li2 If it were possible to control L in such a fashion that we could let L be ideally finite and zero according to imposed control orders, then. 9x rL eo ■ = ex Rl * I# L - o e0 = ®x RL _ 0 % Rl Rl + Lp Rl + Ip L is finite 1.3 It can be shown that inductance can be represented by the physical parameters of the coil and its associated core in the following manner^: L = k ur 1 . 4 where ur = relative permeability of the coil core material; k = a constant peculiar to the coil and core. For derivation see Appendix A. It becomes only necessary to vary ur , and then a method will have been devised for switching L in a two-valued manner^; 6 * E. R. Peck, Electricity and Magnetism, McGraw-Hill Book Co., Inc., 1953, Chapters 7 and 97 7• P. A. Vance, "Saturable Reactors for Load Control, I and II," General Electric Review, Vol. 50, Aug., 1947, pp. 17-21, and Sept., 1947, pp. 42-44. 7 although, generally speaking, L cannot he reduced to exactly zero as was set forth in equation 1 *3 . (1.4.) Method of Varying Inductance Let B q s flux density which would be present in a material if it did not have ferro-magnetic properties. = the flux density due to the intrinsic magnetism of the ferromagnetic material. And define B@ and B^ as, ®o = uo ^ Bi = ui H 1.5 Every ferromagnetic core material has the property B * 3^ 4 Bj^ 1*6 Consideration of the hysteresis loop of ferromagnetic materials will show that these can be represented by what is commonly termed the normal magnetization curve, which is the locus of the end points of minor hysteresis loops. A straight line approximation of the normal curve is shown in Figurei,4 . A close look at the magnetization curve in Figurel^ will show that if a magnetizing force H is applied, it is possible to drive B^ into saturation and then only B 0 increases. 8 eH. F. Storm, Magnetic Amplifiers, John Wiley and Sons, Inc., 1955, PP. 4-o. /5V e%%/%/%%44r4^rr Since it has been shown that, B ■ f B j Bq e U q H then, uH = u0H + Ui % uH = H (u0 + u^) u s u 0 + ui ur • B u0 1.7 and the general expression for ur is Ur s 1 + u0 1.8 Now in the saturated region, Ui = - o H Up s i 1*9 and in the non-saturated state, uo It now becomes apparent that if H can be controlled, then it is possible to switch ur from Up s i 1•11 to ur * l f _ui u 0 1.12 10 As a consequence it follows that _ k L = k ur for air (core is saturated); 1.13 - 1 or L r k ur for core not saturated 1.14 The control of H which controls ur is in turn controlled by- current, since, f _ .kins± I, " -T-- oerstead i where = number of controlling turns on SR; = controlling current in SR; Ife s mean magnetic path of the core in centimeters. Knowing that H can be controlled by a current, H can be driven so that saturates or does not saturate by adding a set of control turns to the original SR and increasing or de creasing as desired. CHAPTER 2 SELF-SATUR/4T1NC- MCHITIC AIvIPLIEIER TRANSFER FDNOTIOE CSolo I Introd'aQtlon . T6 .© aadyted: prosed-ure here will be to develop the transfer fUBOtioa for the hasi© par all el ■= eonne ot ed SH magnet ie amplifierg and to utilize this transfer function as a basis for developing the general transfer fmnotion for the self= saturating magnetie amplifier with reetifler reverse eurrent and magnet 1 © feedhaeko ' '■ Assumptions will be made that (a) the frequency range of the power supply will be such ; : ; that 'the capacity ©ffeots in between winding turns : is negllgibie; - , . - ■ . ' (b) the frequency range of the power supply and the saturable reactor parameters permit negligible saturation reaetanee to exist fThis assumption will generally be valid..for the cas® where the frequency . range of the power supply is held to within 1 0 to • 5 9 kCo and the number of , output turns sf wire do not : ; i become excessive^ } ; . .. , . . - and I©} the diodes present, in the self-saturating amplifier . are not ideal diodes* (2.2.) The Parallel-connected SR Magnetic Ampllfier The basic parallel-connected SR magnetic amplifier can be represented as shown in Figure 2.1. A summation of all the voltages present in the input circuit shown in Figure 2.1 reveals that Si -1/ Ri f 4 r ^ - ~ g ^ io8 at io8 at 2.1 where Nj_ ■ number of input turns; = the input resistance. An integration process over one-half cycle, with time as a divisor, will obtain the average value of the quantities in the above summation, as is shown below: where 4 t = 5 7 ; * a step input of voltage; ii - f iac (denoting a direct current and an alternating current). Taking each term in equation 2.2 and performing the integration gives 13 ■)(' ~ ^p/rs cu/- /?. z. / Z^/^ZZZZ- COA'/Vs?C/~fZ7^S/t' /?/+//?//A/f/? o 7T I ^rr/I/rrr^ < ~ ^P/BJ «-?VV7 C4J --- ►! r— Z/dVX U — r A/^tss-e z.z /7 ) at the end of each half cycle of power supply voltage. (It can be easily shown that equation 2.3c holds for the case of the input resistance assumed zero, and the same general procedure is involved in showing the case in which the re sistance is not zero.) The result of substituting equation 2.3 into equation 2.2 is that Ei = % + % £ £ 1 0 8 ■at 2.4 For the situation where the transient duration is many times the length of at, it is possible to let 4 # - d# ^ t at 2.5 This assumption appears to be approximately valid for even short transients.^ With the assumption, equation 2,4 I'H. F. Storm, Magnetic Amplifiers, John Wiley and Sons. Inc., 1955, P. 1 4 6 . ------15 "becomes *i = TiRi * H ^ , 2 . 6 io8 dt where Ei» ll and # denote the steady state value of the variational components of E^, and # . This equation represents the transfer equation of the parallel-connected SR amplifier circuit, but it would be desirable to express it in terms of and ls0. This means that relationships of the following form must be found; li » fi (I.) = fg (le ) 2,7 f^ (Hq ) can be determined quickly; I; - fl M - R i 2.8 where Kg = voltage gain of the amplifier; E 0 = steady state value of variational component of E@. In order to determine 2?= f% (S*) > the following re lationships will be determined; 16 a. ^ (cosac) b. cos** = (Esr) C. r (S0 ) 2.9 where Eg^ Is the steady state value of the variational voltage across the SR; Cos« is the steady state variational component of cosoc. This set of relationships indicates the following procedure: t 2 (IqI = f3 f" f4 (f5 [s o] )] 2.10 The first function, = f^ (cosoc), can be found by the following process: • » ■ \1 0 ° - dt _ i n8 d ^ - — 0 ^ dt 2 . 1 1 N 0 ® From Figure 2.2, the following integration gives g /cxC n 8 ^ d<4? - i2— Ep0_ / sin wtd^t = <23 + ^ = =2— Ep__ (-cos^t) ^ / -L 2.12 ^ = f ?No %3R (1- 00SoC) Once again from Figure 2.2, - <£?= f <4% = - <#, ^ 1 = ^ 2 * 1 2 17 and by substituting equation 2.13 into equation 2 .1 2 , (1 -cosoc) 3 3 2^fN0 88 2^ s " | S n 0 ^SR (X-oo«) & bOT'ofc - f3 (genTcj 27rfNo 3 2.14 i By determining the average value of e (as shown in k)R Figure 2.3) for a half cycle of power supply frequency, it is possible to obtain cC ^ S R = " ^ S R 003 ,Z = SpSR (1-C 03=c) bR rjj. F — COS cC. SR 7T -7T C08oc = 1 5 ^ - f4 2 *15 In order to determine EgR = f 5 (B0 ), another look at the parallel-connected SR amplifier will be taken. See Figure 2.4. Let IL = + XLb = 23X a ; 2,16 then. Epa = ( ±LI] J R + Egjj + % 2.17 7 7 Solving for and substituting into E0 = IL rL 2.18 18 -T, J L *?#/? jtfe xA— j— x A - £7too _ Trc /7uji JL /v^^zz'd5* ^ 4^ //^ where R = - R^ (in this case only, but in general R represents the total re sistance presented to the flow of current from the power supply during any half cycle of operation). It is now necessary to find the steady state variational components of the average rectified values of the voltage relationship above# Therefore, let dE( Eo - dtT = a BSR 2.20 dBo = A " , V r, dt dt R 2.21 S SR = - Eo = f 5 Wo) Now, by following the operation indicated in equations 2.9 and 2 .1 0 , = fo (cosoc) = 1 0 ^P s R oos°c 2 7T f N 0 20 cosoc = f (ISR) = z 2 L TBs r SpSR -R E ( R t 1 0 8 R 5 , *o] ) 2fN0RL 2.22 Taking the differential of the steady state variational component above, utilizing equation 2 .8 , and substituting into equation 2 .6 , 8 d? - S io a e , ‘ RL 2 fND “ E« ?! = Co) = N, R ^ ^ No % 2f PE° 2.23 Eo Rearranging terms and solving for = 7 in equation 2.23 Ei gives the transfer function for the parallel-connected SR amplifier: Eo _ kE Ei 1 -h p KE T1 2.24 21 N* R where K_ - _i _L = voltage gain of the N 0 Hi parallel-connected SR amplifier; m _ N, R 1 p is the operato]>-^ (2.3*) The Q-eneral Transfer Function Before continuing, it will be necessary to define reference directions for feedback for the three types of magnetic amplifiers under consideration. Since the self- saturating amplifier is actually of chief interest, feedback directions will be defined using it as the reference. When has a positive number, the implication is positive, extrinsic, magnetic feedback for the self-saturating magnetic amplifier only. When hja has a negative number, the implication is positive, extrinsic, magnetic feedback for the series and parallel amplifier types. When hp has a positive number, the implication is negative feedback (imperfect rectifiers always result in rectifier reverse current effects which always give negative feedback.) for the self- saturating amplifier. See Figure 2.5. The foregoing merely serves to indicate that future nota tions will be set up to accommodate the sign convention just prescribed. 22 *SZ-/?/£-J /P/VA7 /?/?/P/?lA£ Aej/Z/ Ac? / C* J cZ ,T' / C* fC'tfcJ t?cJC/\ A ,77 ' — | ^ZT-/ - + A/ A,, - r A ’ T / > v = - A y > ^ A/- - -f- N Ac : //’ccC >>>/- v- d'.y/v os?/y s\? • zv. y //> /Ae'yc* Au& /r 2.S ’ j/ ^o, / ! ■ ’ / + G, 6 Z// // 6v = y^yr fvjpt/S'G’ 2?. 6 &Z<9 2 From servo mechanism theory, equation 2,24 can be represented by a servo block diagram as shown in Figure 2.6. From the black diagram shown in Figure 2.6, a more elaborate servo diagram will be constructed to account for additional factors present in a self-saturating amplifier. The additional factors will be in two categories. Category A: those factors indigenous to the machine. Category B: those factors not indigenous to the machine. If magnetizing current flows and is a part of the load current, = *^L s ™ TL = T Ls ~ (TLxl f TLx2 > 2.25 where Tjc* ■ steady state variational com ponent of the average rectified value of the load current flowing as a re sult of saturation. 2 'Harold Chestnut and Robert W. Mayer, Servomechanisms and Regulating System Design. Volume 1, John Wiley and Sons. Inc. ,1 ^ 1 , "pp. 24 where TTy ■ steady state variational component of the average rectified value of the load current flowing as a result of magnetization; = steady state variational component of voltage output as it appears in the parallel-connected amplifier; K = 1 Tl T ; = S S 2 ^Ls In the case of category A, the portion of the exciting current being transformed into the control circuit of the self-saturating amplifier can be looked upon as feedback. Assume that the portion of which is fed back. Then, •Lk2 = *2L lal RL ^Lx2 % ^ . = ^ 2 = 5£2_ Eol ^ E01 2.26 Rectifier reverse current (as a result of imperfect diodes) might be classed in category A. Here, the current which is fed back into the control circuit is a fraction of the saturation component of load current: 25 U s lr ■ Z — iLs ILs' Rl hp ILs rl kE (^Ls ILs r 1 Sol 2.27 t - where Tkg the part of the saturation load current that is fed back. If a part of the load current is fed back extrinsically to the input circuit, then that which is fed back can be listed under category B: L = ■tim -v -Ls (1 - _ » -v = | t XL — t = =& = [ "V] (1 - ILs R l L % 1 _ W j ILs^b lLa I - :L^L V' kE (X- Efl lol Sfl „ [- & ] (X- i) Soi 4 s 2.28 26 where (-h^) denotes the maintenance of sign convention set forth earlier (i.e., when h^ = -number, then (-hj^) s - number and the feedback is negative, since a positive number will always represent negative feedback by the convention of the servo block diagram being used here)• The necessary information has now been assembled so that a servo block diagram can be constructed (using the parallel- connected SR amplifier as a basis) utilizing the principles of operation of the self-saturating amplifier. See Figure 2.7. Figure 2.7 represents a logical simplification of Figure 2.6. Noting once again that servomechanism theory can be applied to Figure 2.8, Sf0 ^ TlRl = &2 rL _ "pf^ t1 5 1 1 1 i,0iH »>] e (1- j) 1 li >2 * tr-^m (1- ^ 1 f p > 2 ^ hrwhm \ 2.29 The general transfer function for the group of amplifiers concerned (series-connected, parallel-connected, and self- saturating) then becomes 3T " Ar 1 ^ pTc 2.30 %i 27 <>- _ -r _ / yTz zf.7 z!y//9&/7’/?A7 & s C / rsj7c//'& £. 8 •J’//K'y°z/^/lr/7 zr^/?y where Ar = kg (1- &) £ 2 + hT - U- b); T c = k K T i ______ 2 * hr ~ ^ $ ) • Note: Proper choice of functions for the terms in the transfer function must be made for each amplifier case under evaluation* A table showing how this general equation will fit various conditions for the parallel-connected and the self-saturating amplifier is given in Figure 2.9. (2*4.) Modification of the Transfer Function Time Constant For the magnetic amplifiers listed, it is safe to state that (for the most part, at least) one SR is always unsaturated so that transformer action between the input circuit and other circuits in the amplifier can take place. When neither SR is saturated, any change in input current will cause the voltage to divide evenly between the SRfs; and the net result is the same as if one SR were saturated. Figure 2.9 depicts the equivalent circuit. 3 By transformer theory and from Figure 2.10, it is seen that 3"Members of the Staff of the Department of Electrical Engineering, Massachusetts Institute of Technology, Magnetic Circuits and Transformers* John Wiley and Sons, Inc., 19^2, pp. 513-366'• 29 y^W ^yz/fZ /=kZZZZ%f- ZZZZ^^rZ" /?r - O //o '-<~i?f/ /Zs's- yps-r^e*/?/ /4y T <^ * /%» S'<^c Zy J/y z»yy0S's*j c*st 7* £ * & ~~~^ /V* sr-r /7*7 - —/-* >^/r7 ~ / /^O &M Ars*-?J/< sn rrd’//c /v-y’cy7 Acft/c' X^i „ , / £ L - ( « ? - )/ <..*. . 7 ,) JI ' ^ // 77 yr,- " ( / - w ( ' + / * ( £ & * )) z^U"////^ zz/z^kvxr zw//% ^r zzz^^Kz/r zzz 2 % g ^ v r /^z- - <7 /V /?rT? - 0 /Vo (Ojcfs-sr/s/c ^•d‘ Zo z^r y /— «^y / z; z%r ( ^ fz/-- ^^/k%^z7/}yx^ z^^^/xz-T/r />^ /% r////z - zZT^y/KJ/z- zzzz%%% /%fy/7/Z" Z^zy%^c%" ZZZZ/z^yf" » ,* /• v //f-c////^y- s- £~o _ Z f / / ~ <^y / #/? 0 ~ &) /r'±-<^ 7~, A y ^/- / v /?/z^ zr ) ^ ^/4y 'A7 7 O -bJ ' ^ frt+hr ~Y*?(/-ty I >, +/hr/ 30 & / v ' y u r c ? 2 . / o 3^ ~ a/zoa/t* u>o /~ /^ < y /~ e > < ? V / / w /9/7//7&Z' * j’jr / / r-j>77' Z /? /X>_ 2.32 Forming the time constant (where is no longer R^, but has the new value of R Q(^ by the reasoning which resulted, in equa- tion 2 .3 1 ) gives r m /r L*. y. 2.33 where - /Z. f a \ o . = /K: / T. which means that the general transfer function must be modified to read / z; 2.34 where T 7T <4 - C'/- 32 (2.5.) Ideal Input-output Characteristic for the Self- saturating Magnetio Amplifier The following procedure will develop expressions for 2.35 This is in preparation for evaluation of b and. bz. which appear in the general transfer function already derived# The circuit upon which the ensuing work will be based is shown in Figure 2.11. Figure 2.12 shows in a detailed manner the wave forms to be expected over a power supply cycle of operation of the amplifier, assuming that SRA saturates during the first half cycle, and SRB saturates during the second half cycle# Briefly, a signal voltage across the input circuit sets up flux &>/ which causes to saturate at angleoC# As soon as this occurs, the reactance of SRA drops to approximately zero and flows through one-half of the circuit and through R^, since R%, is common to both circuits# Up to the time of saturation, only magnetizing current flowed in this side of the circuit to supply excitation to SRA and to SRB (assuming low control circuit resistance), The method of analysis to be employed here will assume that magnetizing current ceases to flow when saturation load current flows, with the understanding that the magnetizing current is a part of the saturation load current during saturation. SR/\ : & ol/JTCuf' 2 flr O •< TT cot //? 7i' ^ rs - £~/*/oj J~s'? cc P) \ i$as jPey^t c/ £p£S'~'~xr/7 y fPr/~CS /y a y s re /'z/s y j £,crrr’<»**f */* <5*5-Z - SV*s7cJ//~>fJ yosrPsv / >' = V... 1 ^ss-tc*// /rfc/JSTfP*/siy £ts'-/T&/7'f' /%sysff/st/s>9 fs- Zv/^"Z&<*&/(f&* '/^7 ypre* /VssrcsP/ST^ . /sy&y*£ //roPesfO'e/'/sr •fSP/P //r&Pse&’c/ss? tf/PPP Pfj/ P'PFS?^ Jcsytya/t'Ssrjj S*P&f /7f//Z Cess ^r/77-^/Fs?r//ya yvyr^/i/srTvc yyy^ypPPP /^/p. 34 It is important to note the function of the diodes. The circuit in Figure 2,11 is a bridge self-saturating amplifier. Diodes a are instrumental in achieving the tre mendous advantage in gain (as compared to the parallel- connected amplifier by preventing current flow through the opposite SR when it is not saturated (and its counterpart is saturated) and thus preventing the transformation of load current into the control circuit. Diodes b merely perform the function of completing the bridge circuit. The current occurring during two intervals of operation will be analyzed now: Interval 1: The magnetizing interval Interval 2: The saturation interval Both intervals are indicated in Figure 2,12. For the moment, the assumption will be made that SRA will saturate atoj^oc^ and that SRA is in the magnetizing interval. See Figure 2,13. The equation for B (flux density) is cot 2.36 where -5/ = The equation for i^ (magnetizing current) from Figure 2.13 is / ' A 2.37 where a parameter introduced to accommodate different shaped hysteresis loops; 35 ->j V'tl c 4 + ff-TTA'c, Cut o< /r '/&6//^ K-, 1 U -4. C (m s *** V # o I y£ A. - - - ^ ^ >r r w - £ sL^yc J ’/PTZ /-///p s t lc - the magnetizing current necessary to produce the coercive magnetizing force; ^ = saturation flux density. Substituting equation 2.36 into equation 2.37 produces the magnetizing current necessary to excite SRA: y Z C ( / ( Z o s •- { T o S CAJ £J 2.38 An examination of Figure 2.14 reveals that we can follow essentially the same procedure as above. The equation for B becomes y^7- — cut 2.39 The equation for i^ becomes s t u - & 2.40 and using r 2.41 and substituting equation 2.39 into equation 2*40 will result in the magnetizing current supplied to excite SRB during the time when SRA is not saturated. ^ C / ~ ^ ^ • Ur** In order to determine the average value of the total magnetizing current (over one complete cycle) flowing in the 37 gate circuit of SRA for the half cycle of the power supply frequency, it is necessary to find and A% in Figures 2,13 and 2,14: £ - / ^ / ZXJT/?A3 = -TIl. 277 2~.ZXS/F/3 2.43 Each of the areas is ■oc 'o /% - ^ -t- y^zc ^ +y6iccK: <^j'°c‘ ~ /^ /? = >^ 1^/-/ z^/0 0 1- y*z dZxr ^ — y 'S/s? o - xczc C <^*>7— y^s? £j - ^Czc (^°c “' y~X''X7 c<:) 2.45 Substituting these areas into ITy and I T:T g^g and multiplying by a factor of two to account for SRB carrying magnetizing current on the second half of the cycle gives the total load magnetizing current: ■+y)cx -t- jTx. <^d>_rcxr —<2y^Sssy °<~ -Ta x XX S/?/? 2.46 yTzxJtf/? - -7.x, - -^£ T (z+y)0^ ^az - Z ^ oc ZT L 2.47 - ~y/t'xc ^ ~ ^ ^"j 2.48 38 Having completed the study of the magnetizing interval, the saturation interval will come under surveillance next* The total circuit involved when, say, SRA saturates, is that shown in Figure 2*15, since no current can flow during satura tion of SRA through SRB due to the blocking action of diode a. Taking into account that the saturation load current only flows through each SR during each one-half power supply voltage cycle, the average total saturation current over a cycle must be ir = Tp/Pi { /> c & s c c ) cC 2.49 Now, when oc» o , then 1%, - ITjmR, and it is possible to evaluate ^Ims: m s ~ 2.50 Expressing in terms of ITfnq reveals the final expression for saturation load current. T / s 2.51 The value of IT ______/Fc/ct JCt = 2 2.52 r"x;n flt r - M — W — 1 ***?^s+^cr/trsT / CtJO d*^ cs/r'ar/d/?/ CSS'CtS/7* /cf/y/'t? ?7~y/?s7r/a/v y VO vO 40 where E =, R - /T'a/cx ■+/r7 R^a > Rdb = forward resistance of diodes a and b; Rwo = wiring resistance of SR; Rj^ - load resistance; EppS - peak value of power supply voltage; Eda» Edb = voltage represented in the diode equivalent circuit. The total load current flow is obtained from equations 2.46 and 2.51: (j?-/ yJ<*Z v- <=C ~ y o C a 2.53 where -^c - .■4-7T rfo _^T ^ /T y^z- yz- From Figure 2.14 again, it is evident that J^A- ~ ~~£ . /Zc -7%: - 7$ O C — )/*J/S7 c C 2.54 (2.6.) The Modified Input-output Characteristic The term "modified” is used here to mean correction factors made to equation 2.54 so that more versatile equations result for the input-output characteristic. For example, it would be desirable to be able to include a correction factor 41 for rectifier reverse current effects, ana for extrinsic magnetic feedback. Rectifier reverse current will be studied first. The term "rectifier reverse current" here is taken to mean that current which may flow in a diode (due to diode finite back resistance) contrariwise to normal flow. A check with Figure 2.12 will reveal that (for low con trol circuit resistance) when SRA is carrying magnetizing current, then it is also carrying magnetizing current to supply excitation for SRB up to saturation. However, at saturation, SRA can no longer supply excitation to SRB (since SRA transformer action into the control circuit ceases) and the diode in the SRB circuit path has across it e^g for the interval 7 T If the diode is imperfect at all, and has a finite re verse resistance, a current will flow through SRB and be transformed into the control circuit and result in an equiva lent negative feedback of a sort. From Figure 2.16, considering total rectifier reverse current flow, Cc>£ c/c*j z* = 2.55 _ ^ z xSrtB C / J 7% °(-u>£7r <*<9/cX/^cPtsT^trt/f y&f'srr/sm/f {? £?crf-&^s cY/oa/e- d ^T/zz Se-ym ^/y ^jc^sY oY sost Yos~ o C i& s7ce~///'sr S'oZ/iyjrd' y ^ T ^ y y y '^ Z . / 6 Y Y ffr/Y V S /? /^£'YY~/Yyzr where 3L occurs & t cc^0 so that Kno. ito* = • It is immediately apparent that this expression might be changed (merely by combining equations 2,51 and 2.55) to read as some fractional part of the saturation component of load current: J'/fr = //"/- ~ ^/- / / y ) I ^ / 2.56 where hr * -Z" -Z T l - J 2.57 where [-1^1 =_J> ^ ^ /y: Since rectifier reverse current and extrinsic magnetic feedback current is transformed into the control circuit and becomes a part of 1^, the correction to becomes I^c: ZiC ^ 2.58 The new modified expression for becomes J < ^ 2>59 44 From this point on, reference to 1^ will be taken to mean reference to equation 2 .5 9 . A method now exists for plotting the actual input-output current characteristic of the self-saturating magnetic ampli fier in terms of average rectified values of current. (2.7.) Further Determination of the Transfer Function Variables There are a number of methods by which the transfer func tion variables , and 4 . > could be derived. 1. Since j = , 4 = . , and §t = , it would be X u X j-S X as a simple matter to take the variational component of the ex pression for and li previously derived + (and expressed in equations 2.53 and 2 .5 9 ), and make a direct substitution. 2. A slightly more involved method would be to use the expressions for and to obtain the expression for Ap listed in equation 2 .3 4 , and then obtain j , fa , and ^ by noting the correspondence of form between the two equations. This method would simultaneously present (1) Ar in terms of the angle®c, and (2 ) ^ , 4 , and 4 . Choosing the latter method, and multiplying equation 2.53 by the load resistance, and multiplying equation 2.59 by the input resistance, results in E0 and respectively: — C ^ C K - - J/J -//7 o c j y L/4 /- -2 ~£S /4 z ?7 2.60 45 A plot of these two functions, using one as an abscissa and the other as an ordinate, would result in a plot similar to that shown in Figure 3.1. Ar can be obtained by the following process: £*_ jT< 2.61 This is valid for any point on the steep portion of the curve in Figure 3.1 for which it is desired to evaluate 2.62 Following the action indicated above gives 0 - + hr J-IS - h m II 7T 2.63 Ar becomes according to equation 2.62: U c (/*/'-+ sitcL'-S' /% /lr- \yr 7T z 7u\ ” ^ z J l-rr ^ a \J 7T -f - / /rL T, ~>//7 °C fir ~ Ate fr~ yzoj /— T m / __ /&= W v -Zz5 -4 2 /4v- ^ - ^ x z — /4>P7 / - -7z>r / _ -/zx2. 7 " ^ 72-r 7 / ^ - // ^/-r ^ - &?.)______y^- ('/-£)_____ A /7z- ^ A -A»? ( ~£z) ^ ^ ^ - /4/w (/-£) 2.64 This equation for A r shows the values of - 7 1 > / /v- Z' - Z6c^%? oc z --7 / / — cT<9JgC Tzxi. _ Z ■A. i c 7 7 A = -JO/.STtJ jT/^7 ^2 -+cCj>;r? cCy) 2 ■Azc A = 7 7 jTz/rrj a3~//y CC 2.65 It will be appreciated that these equations appear awkward and clumsy at best. Certainly there would be little hope of using them in design problems as they stand. Chapter 3 will deal with attempts at simplification of these equations and will attempt to formulate a method of design. CHAPTER 3 DEVELOPMENT OF DESIGN EQUATIONS AND A DESIGN PROCEDURE (3*1.) Introduction In Chapter 2 the general transfer function (equation 2 ,3 4 ) developed for a group of magnetic amplifiers was found to be £ -tfr ' 3.1 where Ap = ® ^ ■/ >4A -A/r? (/- £) Then, in order that A r and T c might be evaluated, ex pressions for ^ , and j were obtained for the self-saturat ing magnetic amplifier. Equation 2.65 in Chapter 2 gives -2^x/ z ^tt.c /■¥■ y — yoC ) 7T -Z*./7?3- Cjr/v7c’c - -AAxdAJCC S'Z SLzCAte / / t/CfZ. A 7T A~Zs77J A te. x? V-/z— VcXT ^Jyyyoc)) i r JA.fr>s / 3.2 The immediate problem here will be to simplify A r and T 0 so that these equations will be available in a design pro cedure. 48 An examination of Ar will be carried out first* The above discussion about reference gain and time con stant has assumed normal excitation of the S R ’s concerned so that an examination of the voltage requirements (necessary to carry out normal excitation of an SR) will be necessary. The term normal excitation is used in the same sense as that used in the book, Magnetic Amplifiers« by H. F* Storm,^ and implies that excitation which is necessary to bring an SR just to saturation when no saturation load current flows. (3.2.) Simplified Reference Gain The following expression for Ar can be obtained from equa tion 2.64: Ur + f/~ ^ J ctC 1 + z AccO + y-2Ate (/-ycosac) •SSs? &Z. J 'fTJt/rrj rr oC TY-ZlsTff oCdC A first major difficulty concerns the selection of a normal operational quiescent point on the input-output characteristic curve of the magnetic amplifier. Overcoming this difficulty will result in a considerable simplification, since Ar is now dependent on oC . From Figure 3*1, it would appear that the gain of the amplifier is reasonably linear 1*H. F. Storm, Magnetic Amplifiers, John Wiley and Sons, Inc., 1955, PP* 95-96. 49 / f ^ C ^ / X X ^ F t _ T //■r><< //-ya Ct>y~ C’Cf/^cs; ’fl^XZ/ycj- £o\ -/) < <•*■£7 firs'll~ ** j£cp.fs /?cf£S&/ /X7 0O*/ — c?csff>&/t £Z/?*?S- j' / T y ^ y / V / Y K / W T - ( r % W ^ f ^ / ^ y ^ / L r 7 7 ( r /y/TT/ //a /^yyy? B/pc/r 'b /p b ^ B zY t 50 over a considerable range about . Therefore, it seems logical that an evaluation of equa tion 3.3 at would be reasonable as long as the premise in the sentence above holds. This, then, leads to / — Z-4. tc \J fy — // -~r~ I — 2 yC tc . faaX 7 7 2u m s ^ J 7 7 JTt sr? s _ \y y < > K j / J r A r / JZ-T-ZC - hw fzT iC ( / y y ~ 1 7 ^ ) 4~ Z 'i i . c _ y ) TTTtrrrs \77T^ (tLAL ' T T j r ^ s K y A r \M, /& /?r + - hr n cC-ir //•/•/ms / / J -i/n s 3.4 This equation is certainly much less complicated than it was. However, it still leaves much to be desired. Another possibility for simplification lies in somehow eliminating v. v is a variable originally introduced to accommodate different hysteresis loop configurations. Perhaps a closer examination of the term (Z will lead to a simplification. If /r g- , ( 2 - . S 7 / ) = / 7/^ . and it can be seen that other values of v will bound v = j, the extreme cases being if v s 0 C s - j y y ) = 2 51 if v = 1 , (z-.sry) -/.43 so that the term (2-*57v) does not vary too widely about (2-.57v) as long as I Another factor in this discussion that should not be forgotten, is that the terms above are multiplied by z . TTXtms Generally speaking, Le-- will be a fractional quantity which will have a numerical value in the order of (•!), which means that the weight of the factor containing v is reduced. With this reasoning as a basis, equation 3,5 is obtained: \/ - /.O? Stic 1 L Jims J fir _ / % - w # /j/~ -h jL ■4-ic __ /7/yp I”/— / Dividing through by gives A r 7T_ X'j/rjs — /p77? r.?/7Xu/ns _/ cc= 7P 3.6 Equation 3.6 finally reduces to the form - / 7/ 3.7 52 where l±j , Jkcz*. ; • stLc In order to define an ideal reference gain (a gain function with no feedback of any kind present) let h r - O h/7?= O in equation 3*7* Then, /9rJi = /?//% (.9/7 ¥ - / ) 3.6 where Ar^ is the ideal reference gain, assuming perfect diodes and no extrinsic magnetic feedback. If equation 3*7 is modified to include Arj_, then the reference gain of a self-saturating magnetic amplifier (in terms of a so-called ideal reference gain) becomes equation 3.9 below: _____ ~ / y- |~jr h r lfJ— hrr? where h^ = magnetic feedback; hr = rectifier reverse current; i m3 It should be recalled at this point that this equation is only exact for mid-input-output characteristic operation and for core materials having hysteresis loop characteristics 53 which result in v = However, the equation appears to be approximately correct for all other values of v and for other values of near the point of Equation 3*8 can be modified slightly to give the follow ing form: /?rsL ______- / /-x j ~?J M e ) -b*2 3*10 ^ ^ / (3*3*) Application of the Simplified Reference Gain Function The presence of equations 3.8, 3*9, and 3*10 now permit the construction of an approximate straight line representation of an amplifier input-output characteristic. From equation 3*8, it is readily evident that the con struction shown in Figure 3*2 can be accomplished: /tL - ^ e 11 JZstic /ri - ^ /T7//7 3.12 The characteristic shown in Figure 3*2 is seldom present without some sort of feedback modifying it. In order to handle extrinsic magnetic feedback, the follow ing procedure is necessary: 1. Construct a feedback line with slope . __ 2. Construct the new characteristic by allowing ab « cd for all points on "ef in Figure 3*3* In order to show that the process above is correct, write the equation of line oa and line ef in Figure 3*3* 54 £ ~ o f&pc/6ar£/t/ //S7 O ^ < p ^ /a '/*£/7~- a & T ^ & T ’ *:///9/f>s? (j '// / ? - % / % / ^/x/y Absolute value of equation of line oa: A/ 3.13 /<£ Absolute value of equation of line ef: 3 *14 /frx /9r< The difference between (which is the feedback input voltage) and (which is the amplifier input voltage without feedback) should result in the net value of E^: ^ -I L. I v- _ T ^ s / r L _ zr — — ^7n? — J i m /ft Z?- £«\r^Y- ** /?r* ~ ^ T ' ° /?r^i Az 3.15 It is now possible to show that this expression, derived from the construction above, has exactly the form of equation 3.9 with hp = 0 ; - /?r - /?r /Tf T{-/707^^J • ' / -prr / 3.1b By a similar process of reasoning, a like procedure can be shown to follow for rectifier reverse current feedback. The correct procedure is: J<£ 1. Construct a feedback line with slope z- J?r' 2, Construct the new characteristic (saturation com ponent) by allowing aE = cc[ for all points on ef in Figure 3.4. 3, Graphically add the new characteristic (saturation component) to the magnetizing component of the output. It can now be seen that with a few simple operations the straight line approximation of the input-output characteristic of the self-saturating amplifier can be obtained for the feedback rfm/p/f Y/i?*’ e /"/'c rfsryo/'S'e’/-' g'h/r/'&Cf’^r'SfA’sc 6 /tyc/^/rc-’f/je //-rj^ /7c>^7/c>/7'7/C> £>/7 C'/rf " <^vCZ Zyjpi/r e 3.4 jr/?/?/cz//r y//V£- /r/’y’/?#^//?/?sy#/*/ cys?y&s7yr/?/sr;y ( j ’/s<9tv//y Vi O 57 cases if the case for no feedback is known. (3.4.) Simplified Time Constant From equation 2.34, interest will now center around the expression for n =■ + hr - h** (t-5) 3.17 where Na Rl Nt T, = 5 ' 1 Hu (j -f) Equation 2.65 will give the expressions for and j . Using the same reasoning advanced in section 3.2, evaluation of b and Sz results in -A* / lc ( - ^ (C*r~> °< f 77-Z u v-i ^=21 V = Jf ot- -2- _ a s-^V v=i- /- 3.18 --i where 5 A L c \ __ A l l f y - 1/6/dJ «K j J7tkz a .C 3 f A 30 Ltr* j S 77 JTt-ili 5 3.19 «v* _ZL The simplified version of Tc now appears if equations 3.18 and 3.19 are substituted into equation 3 .1 7 : 3.20 ^ = *■'»71 - 7 7 7 7 hr f - h »r (f-hoi) 58 where f - A u, K- = JVV *<■ q /Vo T 1 - N a K I W* /?L If- This equation (like that for Ar ) is only exact at v = £, °(=il , and is approximate for values near and over the interval 0 ± V ± I y / 7 6 % It now becomes necessary to determine how all of these equations can be made valid by proper selection of a power supply voltage to maintain normal excitation of amplifier SR 1s. (3«5») SR Excitation Requirements A look at equation 2.46 reveals that the magnetizing current of a self-saturating magnetic amplifier is A L C (l+v)~< + \/-< COS °< - * \/S/A/ct^ 77 3.21 and equation 2.51 reveals that the saturation load current is n, - r._, 3-22 The sum of the magnetizing current and the saturation current is the load current: T l = T lk * TLi 3.23 Or in a more detailed form, equation 3.23 becomes jr. = ~ Lc F t \/o< * 1/ j/Vo<~| +Tu* Figure 3.1 was used earlier in connection with the simplification of Ar. Once again noting Figure 3.1, it is 59 noticed that when»<- r, the amplifier load voltage is a minimum; and therefore, the load current is a minimum. Also from equation 3 .2 4 , when ' o( = tr JL - Tlx j 4 = V T l $ — 0 ’ and so only magnetizing current flows at^-y. The demands on a power supply having any impedance at all would be terrific, in that the heaviest demand for current occurs at the moment of change of the power supply voltage from one sign to the other. This is clearly seen by adding equations 2.38 and 2 .4 2 , evaluating them at , and noting their characteristics as shown in Figure 3.5: ■AL Tor = = f A ltoT — 3 Mlc - * Mt.t CoSOjJ- =: <2 A lc (\- V CoSCdtJ °<-TT 3.25 The problem of determining the power supply voltage (if it is assumed that the voltage across any given SR is known) is a rather complicated one when it is assumed that a finite amount of impedance is present. An approximate solution to the problem might follow this form: 1. Find the equation of the SR magnetizing current over a cycle of power supply frequency for (probably the worst case). 2 . Assume a sinusoidal excitation voltage across 60 - 7T +7T cJtycye/y *£’:tL\Sr “ C/~y<~ <^5St* y?i%9epS7f‘’/*>JZ/S7^ tf^v'/'y^VTy /ib Zoo a/ df^y/y*>»y; /~~/^ts/~& 3.S r/^//i/& c?z//7/7zr/y7~ z?^*szrz/c-js7rz//r>/?/y/v& />//?&r/ -77 a +77 co 7 /^(cu tj / j o>a7o7 f C ^ o f J / J f V j ^ l / S < ? J . 6 Z7yF &£?£? S?A/77 F'yZFA^ F the SR gate terminals. 3. Obtain the fundamental component of the Fourier Series of the magnetizing current that the SR ex citation voltage would cause to flow. Call this component F • (since the function for magnetiz ing current is neither even nor odd, sine and cosine 2 terms of the Fourier series will appear. These terms indicate an in-phase and out-phase component comprising the magnetizing current• Therefore, maxi mum current demand will be at approximately 4. Evaluate F ^ f ) at /3S*°. (This will be approximate ly the maximum amount of current the power supply should deliver.) 5* Determine an Im sin cot that will give the required current at cjf- Applying step 1 in the procedure above, j / lc (l- V(OSOJ^) AtToT ~ 1 TOT — ~-JA 3 A lcLC (i- )/ cos futiTT )) — C H- A (OSCdA) for -7T £ Cot £ o 3.26 Making use of steps 2 and 3 results in the application of the Fourier series: F( 2'Ruel 7. Churchill, Fourier Series and Boundary Value Problems, McGraw-Hill Book Co., Inc., 1941, pp. 53-77. bn is zero ifF( ^f) is odd. an is zero if is even. See Figure 3.6. The function to be considered is neither odd nor even (as mentioned previously in step 3 ). Choosing to evaluate only the fundamental components of the Fourier series implies that only the a^ and b^ terms need be considered: F (Cut) 5l/J cl -? Ate-L< f-f . VV cos O jF ia jf SAu /(I- V( 6S0jt)(oSCdF£(of J), =r /(-VC0$Cjf-/J(0S(d£f Substituting equations 3.28 and 3.29 into equation 3.2? and neglecting all terms except a^ and bi presents the ex pression for the fundamental of the Fourier series for mag netizing current: F (tuf-) - a> SMtot + bt Coso/f = S / M C o t - a VAl c (OS djf- 7T 3.30 Following step 4 with the aid of Figure 3.7, the value of Cut=^y° is F(O)t) 3.31 63 - S ^ i _ c . »5VZ7 out" —2l^tL^ £03C4jft 7 7 ~ &^i.c •S'j/-r ou’t" 7T -2 CTGJ «jf 'Z '/’/es *r//r7 C4j/' / -jT/oju J //7 w t / =f&st£c S/>7 cot cufz/SS* ’wfz/ss** — y>(. k €T0%f Ccjj / S S ° / =~/^6SS'£* +3. 7 /KFZZ^yyZ^ cTC/^/P^sVr ^7/V/7 / / O ’ s r //*?/77~/ Now, let a fts = current supplied, by the power supply: yf jp$ — -Z7)• p S $ IM Cot A>t-l3S'o 3.32 The final result is quiokly obtained: X/f j = F (cut) (ut-tfy* l3iT0 A n f/-V + /'Y/y V) ^.7olIff s jFpp s — A lc (3 > S'S / «? \/ J 3.33 where v depends on the type of core material to be used. Figure 3.8 depicts load current flow when the load current is only magnetizing current• Writing a loop equation for peak values of voltage for one-half cycle of operation gives - Tf,, s R + L rs« -h Eda. + E db //5 3.34 and solving for results in -/f/r - EfJ*S- Ed CL - Ed b — JT/zy, r R 3.35 where = peak value of voltage across SR; E ■ E >‘>'S - E i c - Edl, j R — Rd* + RJi, -t- dwo ~i~ E l j E^a,Edb » voltages aiding in representation of diode equivalent circuit; 65 Cl S<** £Z't>a/e’ A ^.4*/rz*A?^f /V/fZ/T S r s / i F 0/ ^ ^< ? ^ & r 7*r6/SZ&/JS 0 / ^ / 7 / ? TytP/y /&/? r/Zf &/r/Z7gz'^rz/r*S/?Z' o ncjtnr rj - ^Af/r ^5* //*z Ccf X /yyz/r'z* j.9 /^zt/s /zz-zfXf/?/?y zv z ? /Zz’/r’/y/zz jFxz/r/zr/iZA/^ z>/\ 66 Rda> Rdb = forward resistance of diodes a and diodes b; R^o s resistance of wire in gate circuit; Rz as air core reactance of the SR. Previously it was assumed that the voltage across the SR (when only magnetizing current flowed) was known. It will be necessary as a final step to derive an expression for this voltage. A very fundamental relationship is ** *0 * dt 3.36 Rearranging and integrating equation 3*36 completes the description of the SR excitation: n Vs = / ^ 4? f i m . !*— / s i n « t-St,? -scf, =10- (-c.s*t) - tf’s,, to' Oi No / *77fNo 7T { No r __ P7T Z1 /Vt Ps A-fz tf,K------3.37 where (fs — A-fz j The assumption here is that normal excitation is employed. See Figure 3.9. (3.6.) A Sample Design Procedure In setting up any design criteria, it is desirable to point out that there is no one particular method of design. Usually tjiere are a amfeer of appr©aeh@s depending upon what : is given and what is desired0 The design procedure presented her# is 'used only as an illustration^ It assumes that; the following conditions exist or are desired: ;/ / -io is knowni v ;' '' i ' ^ - • 2 0 Tg :1 s hmoWno: - , ; r ' ■ v ;v': . 'i t / 3 ® Oore material t© he used' is known, / : , ’ ' , Vv ko Peak value of the sinusoidal voltage (input - . : _ -r : signal) is knowno (It is also assumed that . the peak current which ©an he drawn from the: 't ■ voltage souree is knownoi. ^ v v ;1 ' - ' ' ." 5'o Diodes to he used in the ©ireuit are known« : The proeedure that might be followed for a situation such as this is now enumerated0 ' ' : ‘ ' lo' Deoide on desired ^ _ ■ where h^i i® ' the reference gain ohtaimahle from ;' a self-saturating magnetic amplifier with - , , perfect diodes and. no extrinsic feedback; ' Ar is the reference gain fgain present for 1 . a direct current signal) <> ■ . 26 ' Gonstruot a tentative input^oulput characteristic : corresponding^&Ui ehoB@n ahove« See Section 3o3 - . • : : and I'igmr© 3® 10 ® x V : ^ : ’ ■ ' - 3o. Knowing S 0 0 and min. from step 2S compute t 6 -. Prom equation 1@7S ,; . ■ 68 JT0 S T 7 & X . crs7c/ srr//7 . f £ e » / X/psr*/tv's// A 3 J O SAfo/ftF /tyj/7SJ/&// //./. /yi/z^yz- - < 3 / / r / > / / r Z/Y/?///?/ /A/rVJ r/C T// P///-'A*S~ — z/7 //? / r// Eo min. 3.38 4. Find hr from the diodes to be used. Determine values for Eda> Eab» Rda, Rdb- From equation 2.56, ^ = f'tgg r »i 5 ' 3.39 5. Find Kg. From equation 3.8, /• 7/ (.in V-i ) 3.40 6 . Compute h^ necessary to give required gain (Ar). From equation 3.9, »r = 3.41 It is possible to solve for ]%. 3.42 7. Determine Tj. a. Choose > Kg and choose T c * slightly less than T c to allow a margin of safety. b. Compute T^* From equation 3.20 the expression for T c is C ' ^ . (gIQ + hr f - hm (t-I.Ol)) T(. > / 3.44 70 8 . uhoose a value of by using the relationship /TV > TncK.'K, x% <*-X • v 3.45 9. Compute a value for the variable M by substituting a value for f so that a reasonable value for results in step 1 0 . From equation 2.24, Na /f. ke — /Vo zf/ y T/ = ^ /Vo /Pi J f o I A 3.4o Solving for — ■ in one equation, and substituting into the other results in AY/C = i Z L f /ft1 /r* 3.47 10. Compute a value for R^. From equation 3.47 above, the following is obtained: /tV A* = /V/ - fa +£*■) l?L* ^L7- 3.48 where 7? =• +ku)o)+f?L. (%■) * 3.49 Choose Ra > to account for wire resistance (^wo^ * 11. Compute i^Q. 71 Evaluating equation 3.24 for »< = 77 and multiplying by El , ^ A l ( L «— E~Q f H , /z-c — Eoy*'” . - 3.50 12. Compute ITjTiq. Evaluating equation 3.24 for — o and. multiplying by Rl , %/. l*t s tft- ~ ^‘ ^ °'X ZTl-hs = ^ 3.51 1 3 . Compute the ratio of . From equation 3.46, /Va' Kb #a' N» “ 3.52 14. Compute Epps. From equation 3.35, Brs^ e - T „ , « 3 .53 where E - -EJll Substituting the expression for E and solving for EppS , A * ^ ^ ^ 3.54 Finally, if equation 2.50 is substituted into equation 3 .5 1 , then, JTz>i s f t + +’ E<£1> J 3.55 72 where R = -h f/fl Choose R — FA*.*/?d£ V- FL to account for F?wo . 15* Compute IppS. From equation 3*33> 2 pps ~ 'Ci.'. (j-s'S'-h3 i/J 3*56 where v depends on the type of core material to be used. 16. Compute From equation 3*35, Eps* - £ - :r/p5 * ^ 57 where E = Epp* - FA* -EJtb] R s R A* ~t~ Rd b+lfMO+Rt-* 17. Choose a value of e that will make Ife practical. From equation 3*37, = as,£j,Aft N. h r 3.58 From Figure 2.13, H e iVe A L c - • in No 3,59 Solving for Afe and lfe respectively, and multiply- 116 ^ w if«-. ;______’PS* /lfe - ^ B} & / ^ 3.60 18. Compute the number of SR gate turns. From equation 3*59, Ai? He /\/a = .iTTAue 3*61 73 19. Compute the number of SR input turns. From equation 3.52, 3.62 20. Now determine the approximate wire resistance for the input circuit and the load circuit. It will be assumed that the turns and core together will give a cross-section which appears like that shown in Figure 3.11. From Figure 3.11, the desired value for the wire resistance is set forth as follows: The mean radius for the loops is ^ = s' + i + — 3.63 where The path represented by a loop of wire is D^: - 2 jt (K j- J - ir ) 3.64 The path represented by N loops of wire is D: 0 - ND) •= it N (k + fruj) 3*65 Assuming that the input turns and output turns are wound in a similar manner with the same size wire, . z . /Tm/x *=3 7? Nx (° (h i $ Jhw) f?W6 - TT No (° ( h + 3.66 where t ^ , t^i = diameter of wire (subscripts refer to output and input windings respectively); 74 where 10 , * number of layers of wire (sub scripts refer to output and in put windings respectively); h = width of core tape; - number of input turns; = resistance per unit length of wire. 21. Determine the feedback turns necessary and compute resistance of feedback winding. From equation 2.57 (with only a slight modification), A'A/f = N o 3.6? where Nf = fixed number of feedback turns; K = variable multiplying factor. In the case where a number of layers of the same size wire may be wound on top of the input layers (or the output layers) as a feedback winding, * W - N - f CzJtA't'wi v £ uj-fJtf ) T 3.68 where 1^ = number of loops of input wire underneath the feedback wire; tyyl s diameter of input wire; tyjf = diameter of feedback wire; If ■ number of loops of feedback wire. If feedback for the bridge self-saturating magnetic amplifier is obtained as illustrated in Figure 3.12, then, ^ - ^ 7 7 , 3.69 75 j-^z/^- ^r/?ro'/7/?77/y<5 /&&&/>/zr7y and solving for R^, ! - K _ _ 3.70 |j- wj /Vft where K = N f 22. Compute Rf . From Figure 3.12, /?+ - /f, y-zfi. /f, K l /fi ^ e yj 23. The following checks should be made: a. Check that R^ chosen in step 7 meets the re quirement that R j — R w a f fp 3*72 where R^^ = input wire resistance; Rg = generator resistance. b. Check that R Q chosen in step 10 meets the requirement that /Fk - RJ l<( f RA h tRwo 3.73 c. Check that R chosen in step 9 meets the re quirement that R > R 4 *-h RJO, 3.74 ■ : \ ' : 4: - ; ■ . : ' -f ; ■ ■ : "■ w m j m B ■ ■■ C4 olo) . Intreduction A ■ Zt is generally desirable to cheek equations thr©ugh ex perimental results if this is at all possibleo ' In order t© cheek the design equations derived in Chapter 3 9 a sample design will he carried out using the procedure outlined in Section 3 060 Zt was decided to use Supermalloy as the core material and to run input=output voltage characteristic curves for a number of different diodes in order to mete, variation in amplifier characteristicsQ Before undergoing d@sigma preliminary tests were conducted fto determine characteristics necessary for design) on a 0 core material$ and ho diodeso ' f4o3o) Ixuerimental Test Results on Gore Material fhe principle reason for choosing 'Supermalloy (a member of the Permalloy type of alloy), was that a large core of 2 mil tape was available as a supply for winding smaller cores (Small cores are necessary to reduce hysteresis losses at high frequencieso)i Great care was exercised in removing the tape windings 78 since Supermalloy is exceedingly strain sensitive. Even so, there is no doubt that the Supermalloy characteristics were probably altered from the manufacturer's specifications in the process. However, it seemed that as long as altered characteristics were not too damaging, the tape could still be used, since complete data (on the same cores used in the SB's to be used in the magnetic amplifier) was taken. A method (similar to that mentioned by H. W. Lord in an article on dynamic hysteresis loops-*-) was employed to measure the dynamic hysteresis loop of the tape. Figure 4.1 depicts the measuring set-up. This set-up permits an actual visual indication of the dynamic hysteresis loop on the oscilloscope face so that it may be photographed if desired. Calibration is achieved in the following manner. If R jl remains small and the excitation source has essentially zero impedance, then e^ (Figure 4.1) is directly proportional to the core magnetizing current and to H (mag netizing force). Let x be the number of centimeters deflection on the oscilloscope face. Then, Ef,p s A k = x 4.1 1'H. W. Lord, "Dynamic Hysteresis Loops of Several Core Materials Employed in Magnetic Amplifiers," AXES Transactions, Vol. 72, Pt. 1, 1953, PP. 85-88. 79 _ _ /rrfc'Cfs'cr T^os- I 1 rrr l/rra/t?/- /iej/ ye-rf/ca/ /syx* f yo//cKp //& S-/X0 /7S 4^./ /VfrM /Z <7/i/£~/?J'Z//y/A/<5 r//£ £7/^/?/#/ T^'Z/o/y Zc*//csjfC* dfftfoh'yjfljr yo/yctyc dSe r'^/typ&CtZ M: c/c,yos'S77*S70 y>/A^d’ dPcs^ectZ-1 ^/irvST? dZcr*~C> />fa/t^cco/ A / r , £S/7a/ f c /%/kZ7 80 where s voltage gain from input of horizontal amplifier to oscilloscope face; Eppg - peak value of excitation voltage for oscilloscope calibration. The gain for horizontal deflection (for calibration) becomes * £ ft/t s A* 2 Since e^ is proportional to magnetizing current, 4.3 A method is now apparent for determining H: . V7T N a (Ah) /fe 4.4 u .HJTNjlK it,** Ah 415 where A^ = __1_ _ — ^ — \[T A proper choice of and C results in an integrating circuit From equation 3.37, sir 'F fisAfe to * Changing notation so that results in the following equation: B = M y 3 n f N aA-fz 4 , 7 81 For calibration let 4.8 where Av = voltage gain from input to integrator to oscilloscope face; y = number of centimeters deflection. Also, 4.9 Substitution of equation 4.9 into equation 4.7 gives the calibration equation for B (flux density) as iTT-f1 Af e N o /) v 4.10 where A_ = Figure 4.3 shows photographs of hysteresis loops taken at different frequencies. Note that saturation is not clear cut. The process of determining what region of operation to call saturation is difficult. Another method of measuring was used to determine the idealized parallelogram-shaped hysteresis loop used in the design, e^g in Figure 4.2 was obtained directly on a cali brated oscilloscope. A measurement of (Figure 4.2) was also recorded, and through the use of equations 4.4 and 4.7 H s and B g were obtained. The slope of the parallelogram approximation of the hysteresis loop for Supermalloy indicates the approximate differential permeability. See Figure 4.4. This slope was used to plot differential permeability (u^) versus frequency. M a m 3 9 £<3A1C. I l l l l l ■■■■■■ /7-£~rs/-r.J-#: S.4Z ^f^r/fa 4 . 3 0 ^ zzr/w/r/d’ /yxsrs/r’sj'/s / r ^ e r o rr cot zjf /7yA//9A7/c yyrj-T PA/ jy/opps /?7~ /? s^/pp-pt/pyycy tfp' s o src • . ■ •. - ' ■ 8% See Figure 4o60 Figure 7 illmstrates tew tlie dynamie hysteresis lo©p - • ■becomes larger (indicating greater energy less) as the fre quency of operation increases from 30 i ke0 to 40 kc» Figtire 4oS illustrates the results of a test ran on two cores having the. sam,e cross sectional are (A^,Q) and the same mean magnetic path length (l^g/o These two cores were used in the experi mental bridge self“Saturating magnetic amplifierD C4®3o) Experimental Test Results on Diodes - . Three types of diodes were testedj all tests being run on the diodes at the frequency contemplated-for magnetic amplifier operation0 The three types of diodes will be called diodes,and 6= They are classified as followsr Diode A General Electric 1N334 Diode B Hughes 1D6006 \ . Diode G Sylvania IIM38 (Diodes B and 0 were operated slightly above the manufacturer9 s rating when used in the amplifier designed latero) ' The circuit set up for running the tests is illustrated in Figure 4o5 =- By reading and from a calibrated scope3 it is possible to obtain a plot of % versus where - B ■ - . p r s c the peak value @f the forward current ■ ^ R ■ . 'through the diodeo Having made this plot s the problem of determining the diode equivalent circuit was relatively simpleg since and could be easily determined a #ee Figure 4 c, 9® i ' KSH 70 SO zo so so £90 J0O SOO Zv^vf"pt/SA/s y (o/ozss zy y sssoa'sJ /~/<^7 s s 's > 4.6> yo'S’/s'r/s/V o>s's7/sss/FJr//r/s7Z /^^s/vzF^/z/ry^d/J j^/r// /=r/r’ Z0 3-0 Jd/Tc I Z3j — - fT/ > /^ #c= 0.7 oerj *<*^<3/ /~r/j^0'Z'& 4-. 7 /7//l/^A7/Z //XSTS/FSJ-/J /<2&/U (f&''c9//X/'s'7? AT/*/£>,-? ) 87 - 4 ^ X / < d 7 £*/a3 m > ¥c- 4 6 7 \ ^ ^/^&r&//p-/c>^s-&'rr7 /%/o0s~0X''??arf>0S7j fr/7 d p/~ - /Ft/ u r i p /F/oafe I I £ / /Ctc/c* y r^£*si' 7T c o t 27T }Vav4/- //p/ Z V ^ ^ /Z ^ 4 ./0 AfsfT//0/7 /7^7^F/FA/Z/V/F/cZ /?jF/F/r/PcZf y/p/z/A' in the form illustrated in Figure 4ol0. Also3 note Figure 4ol2 which shows photographs of this-; reverse-: emrrent phenomenon, for the three diodes, eoneernedo (In this chapter^ when reverse .current.is mentioneda inference will be to the form just mentioned and not that due to a finite reverse resistance<>) This reverse current phenomenon appeared to be caused by effects directly ©oneerned with the diode<> Xt was more pro- - nounced" with the & diodess less pronounced with the B diodes/ and least pronounced with the 0 diodes* ; . Notes Since a fairly simple method of handling this ■ effect-“by treating it in exactly the ;same way as rectifier reverse current due to finite reverse resistance-seemed to work satisfactori^ ; ;:; . ■ .ly5 a further investigation'was not madeo' Xt seemed that to continue investigation would involve a study of component s 3 which is not the purpose hereb 5 . ' " Since any capacitive coupling network can be made to filter ; ' ' '■ : . . - ":V 2 out the average rectified value of any complicated, waveform , ^ 0T 0 ho Martin3 Advanced Electronic Gircuits, Prentice- Eallg Xnco9- 1955<> PPo ■b09-6iio - o : . .- a s ' /•& y.y z.o /c X 4 . // /v/rsv/?/?# sKrj'/j’7?7A' 0 / 0 0 S /? /A/JS 4- / 7 / o a s N P 60 0 6 / 7 Z O / 7 S C *fy/r*rn/e* /A/ 3 0 t u t - 4 jO/s-ct://£>/? a / a> f Si /Srasr? rrasymr/ A '/&6//'<0’ 4./Z A’/VO TO <&/?/?AV/if ^//O’/V/A/eS /Trsc r / / r/3rsr’ /70-j/^/6S^ coj/f’/T’s / s r 92 a method for determining the average rectified value of the reverse current is immediately apparent• Figure 4.10 illus trates the principle involved. If Ej is the difference in voltage between Epr and the level of the peak after the waveform has passed through an RC coupling network, then, £ r - . R R t __ £ J t t R ~ K 4.11 where 1 ^ is the average rectified value of rectifier reverse current; is the average rectified value of the diode 7T R forward current; is the direct current present in the original R waveform. Merely by reading Ej and Epr from a calibrated oscillo scope and knowing R, then, it is possible to determine IR r * From this information, a plot of I^r versus 1-^. can be made. I-,, is used here to signify the average rectified value JL u of load current flow which has a waveform which is purely sinusoidal. The average rectified value of saturation load current, for example, (which flows in a magnetic amplifier) does not have a pure sinusoidal waveform except at = o • For purposes of design, the following procedure was found to be quite satisfactory. If the maximum saturation load current of the amplifier to be designed is known, then the plot 93 (for the one point only) accurately represents rectifier reverse current. (However, there is a question as to whether other points on the curve accurately represent rectifier reverse current in the amplifier, because of the waveform situation mentioned in the previous paragraph.) Very good design results were obtained by drawing a straight line through the approximate point of maximum load current and using this information to determine hr. See Figure 4.13. Figure 4*14 shows actual plots (of rectifier reverse current versus 1^) taken on diodes A and B. Diode C was neglected because it exhibited only a very small amount of reverse current. Using equation 2.56, V = - N t r 4.12 /Yo (4*4.) Design of a Bridge Self-saturating Magnetic Amplifier The following information is necessary: 1. A p = 150 2. fc > 30 cps. 3* The core material is Supermalloy 2 mil tape. 4. The peak value of the sinusoidal input signal is 2 millivolts, and the peak value of current which the signal source can supply is 2 ma. 5. The diodes to be used will be the A diodes for diodes a and the B diodes for diodes b. The following is obtained by carrying out the procedure 94 fVef/eo/oj / / z. ^ Z7 Z2f^Z" & / rj y A / /-^ $ b Af v~ — — /z /(7 h r <^ H I r ir 4/ 1 < 1 y z 4 z / Otss’/'C'f? Z -Zz^z (/*///// /v &£//'& 4-.Z4 ^ssr/f/f/r / w o /7/scJcf / ? £ A ' r yyy^o- (^ 6^c?crs- 1. Let Ap^ = 300, where Arj^ ;> Ar . 2. Construction of a tentative input-output character istic (straight line approximation) results in E o vtt* =-. y tfotrs — J.S'/OLTS See line ef in Figure 4.15. 3 . ^ = - dL?-jr) — y, ?y 4. From Figure 4.14 and equation 2.56, hr = .211 This modifies line ef and gives line eg in Figure 4.15. From Figure 4.11, Eaa = Eab = 1 TOlt Rda = 5 ohms R,b - 10 ohms 5. K = fir/______J to E " I'TI (.int-U I.Tl((-V7j(g.7‘l)-IJ-'9S:0 6 • \ = (.«,)(*.71)-( m - ‘) h * * This modifies line eg to give line EJ in Figure 4.15. 7 T - ( f - h o l ) fg,7i)-.IS-g ('(z.JLi)-/-Ocl), 3) 1 " ^ r 71 = TT*------< J 77 ~ 3, X JO ~ ^ where it is assumed K = K_ = 25; ®q ^ T . 1 is chosen less than T c = f =S.3IX/o C u 3/7 fc seconds; T c 1 = 4.8 x 10 seconds. 96 o Z0 Z //VS /?/c'/p/70J: //ysf^y^AS ^ 0sh/?/f’/y0/~Z'rf/j 7~/c. (/V/fA stf/sc /^c-c/d&eAJ ( W/ y/r A /& /rfsrjr/r^f/c /^C* t? c/ £ & cT/£ J ' j u r e 4 - J S *fr/r’/?S<$#r / z ^ f /9P/?/?<7X/Af/?T/0A' 0//S7/?/? tTTX'Sr’/j- T/C 97 S. p. > jx/o'3 . n i — ^ = :---— = / A*^AC *2 X / 0 3 Therefore, choose = 1.2 ohms. 9. TVT - 2T-, f ^ 3 (9 - s x > o ~ s){3ox;to'3) ^ t0c n ~ /rr where f is chosen at 30 kc. 10. RT - * [ f e T f — r — 37.S~Jl*nS L X m ' V I W /W where R& is chosen to be Rda - R^-^ = 15 ohms AV _ A <3 -7.9Y ( S k ) = « • 6 E l J h - — 26% M .04 7/ 11. - ~nle~ - -'-y - - H-f /Kick . 12. I_ - - 121 M<< Ims - ^ - ^7r 13* jVL = Af A' = R£./l ‘Z) ^j.oi No /ft -? 7 <»' = JL- t sjo, tEJl - iL. (m'xio-iJfrj.s-J+i-hi =/4,ys-y. 14. Epps where R is chosen to be = 5~ttotH.E = iJ.So> 15. Ipps = (J.ST + J/) - lH.£XID-?f2,SS+l) = CC- 0 ■?** . 16. EpSR = E “ IppgR = Jb.H£ - C6xID'3Cil'S) = s '6i v. 17. Here it will be assumed that Supermalloy 2 mil tape is available. Tape width (h) is .125” If 4 turns are used, Afe = t.i S X /o'3 o* 3' f l A n E ft* IDg | , _ /.z(.clls)(f.6H)to* ) / ^3 0^0^ 4 e = y ^ e Js. V ’JWJ y^)^sjJw ^-s 98 Bs and Hc were obtained from Figure 4*7 18. N0 = =r l?.s'Tunis .+/[ A LC ,1-TT (.0! H5 ) 19* ^k'e fta ) ^ *5" O.J )J q# c- - JO 7 turn 5 For convenience choose N0 = 100 turns; * 100 turns. 20. R^i - ^77 N ^ r f k i - U w ) ^ C . s v ( i o o ) a - » J y / o - 3)(.iJ^i-il'-ois>i)) where h = .125" (See step 17 above.) f> = 6.82 x 10“J ohms/in. (from wire tables^ for B & S gauge No. 29 copper wire) tw = .0122" (from wire tables^ for B & S gauge No. 29 enameled copper wire); J? is assumed to be approximately 3 layers. 21. K /V> = /U M, - fjrsJOooJ = ts~.? ~ 2 7T (^N-f ( h + f'h/a t f) ■= & (C-ZlX/O'2) fcoj f j j f t «? OJC.ouiji- — , 2~J » k nn I where N^ is assumed to be a constant 30 turns and K will be the variable. K = = ■J£JJ1£±1 = e N-f- Jo Ri = = 0 ) f = /.// i - K l-.fJC where Eg is assumed to be 1 ohm. ^"Federal Telephone and Radio Corporation, Reference Data for Radio Engineers. Third Edition, Federal Telephone and Radio Corporation, 1949, P* 41. 4"Ibid., p. 74. 99 (For circuit diagram showing and Rg, see Figure 3.12.) 22# R^. s RjL + Rg ss 1.11 ■his 2.11 ohms It is assumed that R% includes the resistance R ^ . 23# a. Check that > Rwx-htf-fr Rj[ ae 1,2 ohms (chosen in step 8) Rwl= .692 ohms (determined in step 20) s generator resistance - .25 ohms (assumed) 1 *2 > . 6*^0? i~, Ji = - ^ ^ b. Check that Ra > RJL^i-ffJLb+Rwo Ra s 15 ohms (chosen in step 10) Rda = 5 ohms (see step 4) Rdb = 10 ohms (see Step 4) R^0 r .346 ohms (see step 20) 15 — / f A f M /-/Two — 1*^/6 Z? IS.lit c# Check that R ~ /fA i~ ^4 bt ^ 0 R = 42.5 ohms (chosen in step 14) Rda “ 5.0 ohms (see step 4) Rdt, - 10,0 ohms (see step 4) Rwo - .346 ohms (see step 20) R jj - 27.5 ohms (see step 10) 42.5 ~ Rct*-hZ4ti-/r»dttfL - r+loi- 17-S= It would now be of interest to compute R e^ and and determine the actual cut-off frequency of the amplifier just designed. UWv. of Arizona Library 100 From equation 2•31, From equation 2*33, ^ 6 = i t t From equation 2.24, /\Ja L - — ____ =■ J.Of/o-f 1 ‘ 777 /f/. yoo -?7*^ ifcox/o2) Finally, from equation 3*20, r ______[______- ?i'iV.cxio's)z.7i & .(Sc t hr f - h* ( oe)) l.^ll T c = 7, J/ X70" 3 30 that = (c.ti) The amplifier just designed was constructed as shown in the schematic of Figure 3.12. An interesting and informative plot would be one of the amplifier input-output characteristic (without magnetic feed back) obtained from equation 2.60 and compared to the experi mental curve in Figure 4.15: £ c =4^ +wc.sc<-iv 4 T l*, ( L t £ J S ) J 4.13 fV = j~, /fx A (“t- vw°{)fhr Ii.$ t 4.14 Allowing h^ to be zero and substituting values into equations 4*13 and 4.14, ✓- / 4.15 101 where iL_ = = ,oiii •iVN* ,in (ioo) Ijinfi = ~ ( fys-Ffci ~E ^f_J from equation 3.55; ILC1S = , CJ? ( J l L vv. 7 / R = A =s-+lo+ .3si-J7,y= iJ.1 oh», E s .Vl^(3ci + ^CtSM-^SIfla() T 3.1 i ( j + j g l s i ) 4 . 1 6 o Similarly, E4 = — ll.io) (-■±111. (-<-v 3i^)f(‘ii)fiac)(-l±^L)j J o o \ 17 / k= / 4.17 = -.OGSiS-C*- -h-031 ( l ± M l ± ) Ei v 4.18 i/= / If a plot of equations 4.16 and 4.18 is made, the results appear as shown in Figure 4.16. Figure 4.17 compares the design frequency response with the experimental frequency response of the amplifier. (4.5.) Comparison of Amplifier Experimental Results with Design Results By modifying the design of the amplifier to accommodate the different characteristics of diodes B and C, it was possible to run input-output characteristics experimentally and compare them with the straight line characteristics and with the characteristics obtained from equations 4.13 and 4.14. Figure 4.18 illustrates a comparison of experimental re sults (obtained with the amplifier for diodes B used in the - jfx/tf/r/Af^'A' t /?£ / ------2S 2 0 s 20 SO SO 220 JOO fOO / OO /r/?S' S a’/^C/S'er’ 4-./7 J.o /s o S x / p£ ‘/r//*/^'y/r>9Z <0///?/?/? 0y//?/?/?(r7 ~^s?/sj’r/s /^/ra/v ^ £J/~& 4. / 8 Z?/*?/*/ /ZZ f <9 J 2 0 . / A / ^ < L / r ~ &c/7~z?o/ 7~ c“/Z/?/7s? C4o.6o) Discussion of'Discrepancies Between Predicted Input - output ' '' ■ Oharact.eristic. and Ixperimental Results : It is helieved that the first discrepancy mentioned above is a result of the hysteresis loop configuration and power supply loading effectSo : ’ , . ' lifter closer: examination of the parallelogram approxima tion of the hysteresis loop and the actual loop for Supermalloy (shown in Figure 4o4)s> it can he seen that the parallelogram lacks refinement0 Figure 4 o22 illustrates how the actual Super malloy ,hysteresis loop might he approximated using a different straight line, approacho On the other hand3 the photographs in Figure 4® 3 seem to Indicate ho definite region of saturation =. Phis means that when eq.uations prediet maximum output (complete saturation) there is in reality a lack of it0 The second discrepancy also appears to he the result of -Vt? -^0 0 00 <&0 0707?0?/{Z0'7~ 0 / a /s //y/?7~/0A/ 0 ^ <0000'/F/y<070r/f’/J'0/0 Z?/*//?//7. / r;//?//tts? /7f. /y//*{/r- e t / r ^ r oz/srsFocrd-sv/jr/y: fc d /o d e s ) s.s -4 0 ~ZO O ZO <20 //i/z^o'T' y o / T / ? o z (s/;///y*//j ) /9/V/?//y/sF/? Z7. C. /A/s^t/y- /4/X/71f)?LfUy^ yy 110 poor transition between the state of non-saturation and saturation. This indicates that a point chosen midway in the transition range for the parallelogram saturation level would result in a form of over excitation. In an effort to supply as much of the current as possible for proper excitation (through a finite impedance and by using the parallelogram approximation) an attempt was made to deter mine a peak voltage that would supply the peak current demanded by -the fundamental of the Fourier series expression for the magnetizing current waveform. See Section 3*5* The photo graphs in Figure 4.23 show comparison output waveforms of the amplifier for magnetizing current only and for ©< approaching the angle for saturation load current only. (4.7.) General Discussion of Other Experimental Data Suppose it were desired to determine the amount of feed back necessary to increase the gain of the amplifier whose characteristic is shown in Figure 4.19. For infinite gain, construct a feedback line, oT>, as in Figure 4.24. It has been shown by equation 2.57 that 4.19 or from equation 3.52 that h^ can be expressed in a slightly different way as 4.20 4.21 A f p r“ox /m** ic(K J /yZrpS H rfes/Ty /c ? s t / f d^zzy cT/zZ « «_jG? fis t'-a /'/*/? O s * < f . . /K foys? < r//2s /t c ? ^ o a a / ^TlzA/yyyZ A. u/t ------co'f * € £ 7 //'< ?c / / 0 / 7 o / c u t A < * z w eu'/>,7 ^ & c Y / < 7/? ofyeAu/s^s'ap*A / c f's’ocosj. / 4 2 3 /°2/2>r<2 J . s J o I ^ /.s I I m /^c/ t yoA 7y&&Ar (y*?////*' cjryyy?;&///' // m ^f s9/*/*/p&7y?//y' 4 . Z 4 - /?/?/%/ / ys/? /7.c. /A//c* where = fixed number of feedback turns; K = variable (multiplying factor). By using the techniques enumerated in Section 3.3, the new characteristic is obtained. See Figure 4.24. Figure 4.25 illustrates an example when feedback produces slightly less than infinite gain. Figure 4.26 illustrates how the cut-off frequency varies for two different values of input resistance. Diodes A were used in the diode a position of Figure 3.12. The quiescent point was chosen such that ®< 2= 127°. A fairly close check was obtained by using a Bode approximation of the amplifier frequency response obtained from the amplifier transfer function (which included a term for a filter— see Figure 4.27-- placed on the amplifier output for scope presentation), fY = /-?7 0 114 2 0 j"r/T >s?/& //r Z /zA^ r s^/^/i/roA //v.s? 7 '/ o a / ^//,f2rsi/7y72_ y ^ Z ^ t V y ^ y£.Z£T osij- ^-/r/rf^f- 0 /r //i/yp^r /r’£'J-/J' .ss? 7~{/&/?r/&/vr /?/Vczz r / a t / r < ? 4 . z v / v z r f / ? o j - f # tf/vpt/rrt/r r x ^ / ? / / > / s?/*/^//^/*-/? CHAPTER 5 . : : • - ' a o N o m s m m : C 5olo) Limitations am# Adirantages ©f High. Frequency The use of a high frequency power supply in a magnetic amplifier presents a mmber of problems* ' Reverse current flow, eaused by the diode effect (men tioned . in Seetion 4o3) s introduces" a loss' in gain* (This loss ©an be overcome by introducing extrinsic magnetic feed back if the, necessary complication in ©ireuitry ©an be toleratedo ) , ■ ' - : ' i;.--: v " ; ' The ^fattening83 of t he hysteresis loop at high frequen- oies (and the consequent increase in H0) and the diminishing of Ug (differential permeability} all indicate the need, for better core materials® " Another troublesome point eonoerns the attempt here to show the self ^saturating amplifier as a voltage gain deviee«, • A ma^or difficulty is fas with low frequency opera.tion) its relatively high demand for signal current0 However s;" there are indications that these difficulties ©an be reduced by following the same techniques as employed at lower frequencies Thb, chief advantage of increasing the frequency of the power supply is the resulting effect on the time constant» For esamples from equation 3ol6s the following can be obtained 118 T c “ Ket 77 ------. C 3 C +tlir Y - bto* (f-Z.ol) T c 'tlL r JfL — ^/V0/) -9/ .C7£ T .77S fjpLi'JPL _f_ o - * yV. / 10 . 5.1 where K - ^ JY± 'Eq “ /v6 — /v, fix It is seen that, within limits, if f is increased, T c is decreased; and a better cut-off frequency is obtained. High frequency operation results in smaller cores and consequent reduction in size and weight of S R ’s. (5.2.) Other Remarks An indication of the poor results which could be obtained if a design were made on an amplifier assuming perfect diodes and no finite impedance between the power supply and the SR is obtained by attempting an actual design using these assumptions. Results will immediately indicate two discrepancies which are important. As an illustration, a comparison to the design results obtained in Section 4.4 will be given. The amount of magnetic feedback (h^) necessary to pro vide the gain (Ar ) desired was found in Section 4.4 to be If the assumption of perfect diodes (no reverse current and no forward resistance) is used, then the requirement be comes 119 . " ==oOS3 2 : . ’ $hls means ttot M d the second, value l?een used.,, more .negative feed/back would have been applied to an amplifier wMch. already had too mueli and the gain would have been even more remote from that desired. Figure 5ol indieates graphically the results of assuming no reetifier reverse current0 The design value for.the peak voltage of the power supply was found to be ■ • - - ' ■' Epps's 1 0 =45 volts9, : whieh eheeked out experimentally* and yet the assumption Of ;' zero finite impedance between the power supply and the SB would have given ‘ h'-’l ^ ^pps » 7 ° 50 volts e ' • , ' . ; Other effects ;of these two assumptions appeared negligible in the sample design carried out9 but certainly the discrepan cies in the two sets of results illustrated above would be ' sufficient to prevent any accurate designo The diode effect is particularly important9 and more investigation is indicated® 120 JO 1 r„ I.. as -20 o 20 / / 7 y D C / S y < ? / f s ) e's'/sr? ft-rfcr/ JO. /sy& F /7 c Induetanoe of a Toroid Two examples will be used. Consider the first example: A toroid with layers of wire wound in a circular manner. See Figure Appendix.1. It is assumed that the following information is available concerning the given toroid: c is the distance from the center of the toroid to the mean path in the center of the core, a is the distance from the mean magnetic path to the first layer of the winding, b is the distance from the mean magnetic path to the outermost layer of winding, b-a is the thickness of the winding. dl denotes a vector increment of the path which links the current-carrying wire. is the distance from the mean magnetic path to any layer between a and b. By Ampere's circuital law and using the MKS system of units for the present, 1 2 y%r^/7 S7/i?>s~ of £c>/- / — . /7f>peic*/y / j ? , / /? 7&/F&//7 sr/zy/ M tv//F/?' sr&t/A'/z Z Y / ? Z&e?p o Z JV'/S-'C* > * 1 t \---- 6 t '2 Z? TV/FO/ZZ tY/r///7 S/Ate//:- //?/£-/? /7z~ But in a toroid L H ^ ~ 1 > because H and dl are in the same direction and / H i M =■ o Xc - /e 3 After integration, 1 G = 1 7 ? C H 4 for one turn of wire* If there are N total turns of wire, then certainly the path links N loops and /Vr - itt c H 5 If the toroid is considered from the standpoint of flux Jl Q) - & $ Jl ^ n 4. J 7 wh&y'f? yt ~ Uittt /ccf oy f 2"o ^ = f~§ a Jl*. s but since - j § » I (os [ b J I = B C ciL£j ,'i and for the toroid cos/Em = 1 because & is parallel with It is possible, therefore, to write /- /-j /Mr $ / B A 1*. - / y j^ j /5 2 7! C41^ where da = 2 7? p n^ = number of loops per unit radius 124 Integrating the flux gives a- f t 4 j r jW^/Vv.w*- -- 10 Now let - number of layers of windings: Ni = >jji (y-*-) 12 and solving for ^ : 13 and then substituting into equation 5: 14 for a layer of b-a units, each layer having one loop. And so, BN,Nl7T f I 3-4 3) =r ^ ^ ?-( I*- +*& t-* V 3 ( 6-4./ 15 where N^ is the number of layers Ng Is the number of turns in a layer Let N s Nx N2 16 (3?r _ B N V (i'. + .dL+t') 17 J q t = "JL (h' + Jh- 18 For a particular case and for many situations, the 125 assumption of a linear, isotropic, homogeneous core is legitimate: J B = u J H 19 where u = constant. Therefore, J(()r — ^ ^ d ^ 20 3 From equation 5, J. H = J w C 21 Substituting equation 21 into equation 20 gives f m T = " E l i ( t 1 i < U L y - ^ V j j /7C 22 _ A/^M + , dr 6 C 23 It is desirable to express L in terms of ur : W- Iflx/o'7 C{r in MKS system of units. 24 Substituting equation 24 into equation 23 gives N 1 ?/? X. /O'7 *Jr-h4.1) L = 6 c in MKS system of units. 25 In order to maintain dimensions in centimeters, substitute / cz*. - /O'* 'hte tcrs into equation 25: L _ + 0 x n>~ ** Ur N x f +*-&-+clx) _ 3 jj xicT^CjrN1 ( ^ ^ 26 27 126 where K - y- ttx/o'* /V1 f i 2 ± ^ ^ 5 ) ur « 1 for air; up = some other finite value for volumes other than air. The second example under consideration is a toroid with a single layer of wire wound to conform to the core configura tion shown in Figure Appendix .2. The equation for this toroid will be developed in a condensed form since the procedure is identical to the case outlined in the first example. The following will be defined: r^ is the radial distance from the center of the core to the inside surface. r% is the radial distance from the center of the core to the outside surface, b is the width of the core, r^-r^ is the thickness of the core. From equation 6, N i - JLL jin ( r*. ) 77" f#)+ft.) 2 8 where o = ^ (f = B N ( /i-rj I 30 where B - u H dB H 12? for linear, isotropic, and homogeneous materials. From equation 28, j Ne*-1 31 dtp __ N* I-tt Xio~7 Crx-r,) t> ^ . T' (ft + A) T 32 where u = X id'1 tty in MKS system of units Converting to centimeters gives L s ^ k l A r ln oentlmeters 33 **£ ( ^ /L / where K s 1 fK-Kjl Y' + n BIBLIOGRAPHY Books Chestmtg Harold.s and Mayer s Robert W 0 3 SerTomeohanisms and ' Regulating System Designa 1, John Wiley and Sons, In&og Hew Tonk9 1951a Chapter 83 ppG 194-202» GMrchills luel. T* * Pourier Series and Boundary Value Problems„ . EeGraw^Hlll Book Companys Ineog Hew York9 .19418 Chapter m m*: 53-77 > -a ; : Pederal Telephone and Radio Corporation^ Reference Data for Radio Engineers, Third Bdiflons Federal Telephone and Badi© Corporationg Hew York 42 Ho Y 0 s Chapter 3s p® 41; Chapter 5$ p@ 74« - - . - ■ ■Harbin g To L* 3 Advanced Fleotronlo Circuit s8 Prentice “Hall 9 Xnco Hew Yorkg 1955g Chapter 17g"ppo 609-611o Peok9 S 6 la<, Eleotriolty and Magnet 1 am, MoC4raw=-Hill Book Coo3 ■ Ineo g- Hew Yorks 19533 Chapters 7 and 9* Storm? Ho Fo $ Magnetic Amplifiers^ John Wiley and Sons8 Ino, 8 Hew York? I955s Chapter 13 ppo 4-6; Chapter 4S pp.64-65; Chapter 7s PPo 95-96; Chapter 10s p0 1 4 6 0 Articles . - Gang, A. Go <, Electrical Engineering. ;■ ^Application of Thin • . Permalloy Tape in Wide-band Telephone and Pulse ■ ' Transformers/AYolo 6 5 , 19463 p6 177@ ■: Goulds Ho Lo Bog Bleotrlcal Engineering, ^Magnetic Cores of Thin Tape Insulated by CataphoreSlSg* Vol. 6 9 8 1950; ;; PP* 544“548 . - Lordg Ho Wo 0 ALEH Transactions 0 ,?Bynami© Hysteresis Loops of : Several Core Materials-Employed in Magnetic Amplifiers 9 ® • Volo 7 2 a pt0 lg 1953S PP6 8 5 - 8 8 0 Itilesg Jo Go A.IF1 Transactions. wTypes of Magnetic Amplifierss 61 Volo 71, pti I, .1952) PP* 229-38s . ; • ■ . 12 Royer <, Q-o Ho „ AIEE Transact ions,, 55 A Swit oiling Trazislstor D=0 to il“G Converter Haring an Output Frequency Proportional t© tlie D=C Input Voltage,n Volo 74$, pto Is, Julys 1955s pp. 322=326o Vanees P 0 Aa General Flee trie Review, ^Saturable Reactors ■ for Load Control/5 I and 11, Vol. 50s Aug. s 1947» pp. 17-21, and Speto, 1947» PPo 42-44.