ON THE NUMERICAL ANALYSIS OF MAGNETIC AMPLIFIER ClRCUITS Thesis for the Degree of Ph. D‘ MICHIGAN STATE UNIVERSITY BURTON HOWARD WAYNE 41960 This is to certify that the thesis entitled presented by has been accepted towards fulfillment ments for M1 Engineering of the require Ph. D. degree in LIBRARY Michigan State University ABSTRACT ON THE NUMBERICAL ANALYSIS OF MAGNETIC AMPLIFIER CIRCUITS by Burton Howard Wayne In the years 1948—52 a great deal of work was done. by many people to qualitatively explain the operational principles of the magnetic ampli- fiers. Many articles have been written on this subject, During this period the empirical design of magnetic amplifiers was developed extenw sively. However, in recent, most of the published papers have dealt with the application of magnetic amplifiers to specific circuitry and the analysis has been limited to a few people“ In this thesis. a new formulation technique is developed, by con— sidering the core characteristic as a function of the ampere-turns vari- ables. The core characteristic is considered to be the saturation curve, with finite slope, of a ferromagnetic inductor» The formulation pro— cedure in the example yields a pair of first order simultaneous differential equations which are explicit in the derivatives. The coefficients of the derivatives are functions of only the amperemturns variables" A review of linear transformations of variables for two-winding inductors and the extensions to inductors with k-windings is given in Section II. In Sections III and V theorems are developed to give a criterion for the locations of the inductors in the network such that the ampere-turns variable will appear explicitly in the system equations, The arrangement of the elements of terminal and transformation equations, such that they may be written systematically is the subject of Section IV . till/l l Burton Howard Wayne The details of formulating the system equations such that they are adaptable to a computer solution is given by an example, in addition to the computer program used. ON THE NUMERICAL ANALYSIS OF MAGNETIC AMPLIFIER CIRCUIT S BY Burton Howard Wayne A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOC TOR OF PHILOSOPHY Department of Electrical Engineering 1960 ACKNOWLEDGMENTS The author is grateful to Dr. H. E. Koenig, his thesis advisor, for his constant guidance in preparing this thesis. His fundamental work in the area of numerical accuracy forms the background of this thesis. The author wishes to express his thanks to Dr. M. B. Reed and Dr. L. W. VonIersch for their constant c00pera- tion and encouragement. ::< >1: >:< :1: >§< :4: )k >:< >:< >1: :1: ii III. IV. V. VI. VII. TABLE OF CONTENTS . INTRODUCTION ........................ . BACKGROUND ......................... GRAPH EQUATIONS. . . ................... PROPERTIES OF SYSTEMS CONTAINING MULTIWINDING INDUCTORS . . . . . .................... .SUFFICIENCY CRITERIA ................. EXAMPLE .......................... CONCLUSIONS ......................... APPENDIX A ........................... APPENDIX B ........................... BIBLIOGRAPHY ......................... iii. Page 23 27 36 46 49 51 I. INTRODUCTION The term 'magnetic amplifier' was introduced by E. F. W. Alexanderson in 1916.1 Today magnetic amplifiers are found in nearly all branches of industry, especially in the control systems area. Although the magnetic amplifier is relatively simple in its external appearance, the nonlinearity and double-valuedness of the hysteresis loop of the core material makes the analysis of the magnetic amplifier very complex, unless gross approximations are used. Perhaps the . most frequently used approximation is to represent the core character- istic by a series of straight lines, one with infinite slope in the ”unsaturated region, ” and the others with zero slope in the "saturated region. "2’ 3’ 4’ 5’ 6' 7 A serious difficulty of this approximation is that the flux is independent of the current in the unsaturated region, while in the saturated region it is independent of the voltage. Hence to obtain a solution over the entire range of operation requires two different variables which must be interrelated at the intersection of the straight line segments. Milnes and Law8 have analyzed the magnetic amplifier using straight lines with infinite slope in the "unsaturated region" and zero slope in the ”saturated region. " However, only the steady-state component of the solution is discussed, and the analysis is based on linear approximations in the two regions of the. saturation curve. Ettinger9 assumes the core hysteresis loop to be rectangular with infinite and zero slopes. Two distinct variables, one for each mode of operation, are again used. With the advent of the computer, many problems once considered impractical, become practical. The objective of this thesis is to formulate a set of equations describing the magnetic amplifier, such that they are adaptable to computer solution. In following this objective 2. change of variables discussed in Section II has been introduced such that the slope of the B-H curve may be conveniently incorporated into the equations. By applying a change of variable it is possible to establish a set of terminal equations for the inductors in which all coefficients are constant, with the exception of one. Furthermore this coefficient is a function of one variable only, namely the ampere-turns of the core. In formulating the mathematical model of systems containing ferromagnetic inductors, it is desirable to have the ampere-turns of each inductor appear explicitly in the system equations. The objectives of this thesis is to present techniques for formulating the mathematical model in this form and show an example of a numerical solution to a typical magnetic amplifier circuit. In this solution the saturation curve is approximated by segments of straight lines with finite slope. To the author's knowledge the approxi- mation of the saturation curve by more than three straight lines has not been used. However the formulation techniques in this thesis permit the use of a computer such that the saturation curve may now be approxi- mated by any number of straight line segments. II. BACKGROUND 2. 1. Terminal Equations for Two-Winding Inductors. The problem of numerical accuracy for a two-winding ferromag- netic inductor is discussed by Koenig. 1° To review the basic problem, let the terminal equations of a two—winding inductor be written as vl(t) R1. 0 ' i1(t) M11 M12 il(t) = + — (2. 1.1) gm. V2“) 0 R2 ia“) M21 M22 izIt) or symbolically, ‘U(t)= R. Slut) +°Wt§g J’ (t) In the above equations, the coefficients in the matrix 6WD are found from a series of open circuit tests. For high permeability cores such tests are impractical in that acceptable numerical accuracy is difficult to realize. That is, the' determinant of the coefficient matrix W is nearly zero. In order to alleviate this difficulty, let a new set of vari- ables be defined by the following non-singular linear transformations: v?(t) -n.. 1 v.(t) : (2. 1. 2) v3“) 0 1 vat) which may be written symbolically, ‘Unm an, W‘m 1%) -n.. 0 1m = (2. 1. 3) 1%) n12 1 izlt) which may also be written symbolically, Q, “M where n12 = Nl/Nz and n21 = Q1 NI (0 Nz/NI: N1 is the number of turns on winding the number one and N2 is the number of turns on winding number two. The linear transformations on the current and voltage variables have been chosen such that '1 (IV = (On. 1’. Substituting 2. l. 2 and Z. l. 3 into 2.1. 1, a new form of the terminal equations results. n _ v. (t) R2 + nit R, v1:(t)J R2 _(M22'n21Mzi) Definition 2. 1. 1. (nilMll 'nziMzi‘n21M12+Mzzl _n R2 11 (t) .n 5 R2 12(13) ,n (MZZ‘nZlMIZ) 11 (0 F1? (2., 1.4) ,n M22 12 (t1 Leakage Inductance The leakage inductance between the jth and the kth element of a mutually coupled component is defined as ij = M JJ ' “3k Mik' Assuming Mjk = Mkj and applying Definition 2. 1. 1, 2. l. 4 may be written as n .. V1“) R1 + 11.221 R2 v§(t)_ _ .n _ R2 11“) _n R2 12“) _ .n _ L21 11“) d E? n (2.1.5) M22 L 12“) m--- The leakage inductance terms are usually very small compared to M22. In addition one of the new variables is proportional to the total ampere turns of the core, which is of equal importance to the study presented in this thesis. Specifically, $0) = 12m + 111211“)- Terminal Equations for Three—winding Inductors Let the terminal equations of a three-winding inductor be written V1“) V2“) V3“) R1 = 0 0 o 0 R2 0 0 R3 i3(t) i1(t) izit) + M11 M21 M31 M12 M13 11“) M.. M... 3—, 1m M32 M33 is“) (2.2.1) In the study of systems containing inductors With high permeability cores it is desirable to formulate the equations in such a way that the total ampere turns appears as one of the variables. Let the transform- ations on the current and voltage variables in the system equations be defined as: 1‘3“) ‘ 1%) i?(t)_ -v§‘(t)1 vZ‘m _v§‘(t)_ i -n13 0 0 'nza _ I113 “23 F ”‘31 0 0 -n3z ' o o l _ uni izIt) _ ialt)_ V1“)— V2“) I v...) (2.2.2) (2.2.3) Note that the voltage coefficient matrix is the transposed inverse of the current coefficient matrix. The form of the terminal equations 2. Z. 2 and 2.2. 3 into 2. 2.1 is vim v?(t) vim where, uSing the definition of leakage inductance given as Definition 2.1.1, 2 L31 +r1311413 I431 +n§2 (L23 ‘L21) L31 R3 R3 R3+n§1R1 R3 R3+ Hg 2R2 R3 r esulting from sub stituting L32+n§1(L13 ' L12) 2. L32+ r132in L12 .1? 11 (t)— i3“) 1%,) (2. 2. 4) L3. _ ii‘m L32 {i—t 121W Mzzzlzj 51:13:" z _ 2 L31 + n3.1 L13 - M33 ‘ n31 M31+ n31M11 ' n3, M13 2 _ 2 L32 + n32 L23 - M33 ' n32 M32 + 17132 M22 ' “32 M23 2 _ L31 + 1132 (L23 " L21) - M33 " n31 M31 + 11:3an M21 ' nasz 2 _ L32 + n31 (L13 ' L12) - M33 " n32M32 + n32n31 M12 " 1131 M13 For convenience let the (i,j) coefficient in 2. 2.4 be symbolized as Li‘. and write 2. 2.4 as ij’ vim- v?(t) vi‘(t)_ ' n L11 n L21 n _L31 R3 + “$1 R1 R3 R3 n L12 n L22 n L32 R3 R3+n§sz R3 L13 L23 n M33 R3 R3 R3 11%)- i?(t) i?(t)_ 1%) 12%) 13m (2.2.5) Since the mutually coupled component is a bilateral element, i. e. n Mij = Mji- the off-diagonal elements of 2. 2. 5 are equal, i. 6. LEE: Lji” 2. 3. Leakage Inductance Form of Terminal Equations for a k-Winding Inductor. The current and voltage transformation matrices for a k-winding inductor are ii‘m [-nlk o - - o o i1(t) .11 . 12 (t) 0 —nzk - - 0 0 lg“) ifi_1(t) 0 0 - - -nk-1. k1 ik—1(t) DE“) nlk nzk ' ' nk-l,‘ k 1 ik(t) (2° 3° 1) WW) I mm 0 - - o l I v,(t) I 11 V2 (t) 0 'sz ' - 0 1 Vz(t) n vk-,(t) 0 0 - - -nk, k4 1 Vk-1(t) n t o o - - o 1 t j _Vk( ) _ L i ka() (2.3.2) where ik(t), specifically, is proportional to the total ampere—turns of the k-winding inductor. The transformed terminal equations (leakage inductance form) for a k-winding inductor, as determined by substituting 2. 3. l and 2. 3. 2 2“ into the coupled circuit form of the terminal equations, are a): The coupled circuit form of the terminal equations is when the voltages are written as explicit function of the currents. vim l vim v:_l(t) vim where and (Rk+nI<1Ril Rk " Rk - - n n L11 L12 ‘ ' n n L21 L22 ' ' n ' ' ' Lk—l, k-1 n n Lk, - - Lk, k-1 n _ 2 Lnn _ nkn I"Inky-5 + Lkn n Lnj = I-‘kn + nkn (Lnk ‘ Lnj) n Lnj = Lnk n Ljn = ij n vn(t) = -nknvn(t) + vk(t) 11 V1 (1:) = Vk(t) .11 . 1n(t) = 'nnk 1n(t) k- l .n _ . ik(t)— j=21 njkij(t) (Rk+nlz(, k- le- 1) Rk n-le na‘j 71k nf’k, ”1.... nyfk nafk Rk .n . ' 12(‘1 Rk i Rk_ L —.n .. 11(t) .n, 1zit) 112-40 -.n 11“) 1:,(t) 12m * ifim (2.3.3) (2.3.4) (2. 3.5) The self-and mutual-inductance coefficients‘ of any two windings, j and k, can be related by an expression of the form C-M C- M.. M “J nn : Jn J] 2 nj (2 3 6) Z 2 Nn . Nj Nan where‘Fnj cjn = knj and is called the coupling coefficient. The coupling coefficient is less than or equal to one. Hence, if knJ = 1, then Cni: cjn = 1 and ifknj < l, an < 1 and Cjn<1' From 2. 3. 6 and Definition 2. 1. 1 the leakage inductance coefficients in 2. 3.4 can be expressed in the form n Ckn Lnn=Mkk(l+ E:— - ZCkn) nslk nk Lnj‘Mkk( + an ‘ kj' kn) n J (--) n n — 1 Lnk ‘ Mkk( ' an) Since the k-winding inductor is a bilateral element, i. e. Mnj= M- Jn’ the off diagonal elements are equal, i. e. L3, = L; for j 7! n. Under these conditions we see that upon comparing Lin and Lgk that as the co— n efficient of coupling approaches unity, L:k__> 3:331 2 Furthermore, if the core has very high permeability, such as those found in magnetic amplifiers and/or if'the coils are wound on a toroid the coefficient of coupling approaches unity and as a very close approxi- mation LEI}: L:j = 0. Under these conditions the inductance matrix of 2. 3. 3 contains zeros in every position except the Mkk position and the value of the Mkk coefficient is proportional to the slope of the B-H characteristics of the core material. Even when the coefficients of coupling is appreciably less than unity all leakage inductance coefficients can be taken as constants, leaving Mkk as the only nonlinear term. The important advantage of the leakage inductance form is that Mkk is a function of one variable, namely i;(t). The objective of the following 10 section is to establish the requirements sufficient for formulating the system equations to include this variable explicitly. All coefficients in 2. 3. 3, except those in the kth row and column, can be determined from steady- state impedance measurements. The terminal equations in 2. 3. 3 is the more useful and will be used throughout the analysis. Hence the measurement of the coefficients of 2. 3. 3 is discussed briefly. r“ The transformed equation variables must be expressed in terms of measurable voltages and currents, thus the transformed variables should be expressed in terms of the original variables as given in 2. 3. 5. Note that, in 2. 3. 5, except when the terminals n or k are shorted the voltage matrix contains terms of the form (Vk - nkn Vn) for n 91 k. i The term (Vk - nkn Vn) is a very small number and usually cannot be determined accurately by taking the difference between the measured values of Vn and Vk- Also, ifi(t) represents the ampere turns of the ‘ core which can be made zero by Opening all of the windings. It is im- possible to determine any of the coefficients under this condition. However, it is usually sufficiently accurate to assume the exciting current is zero under short-circuit conditions, provided that this term is not multiplied by a large impedance. The short-circuit condition also removes the small difference voltage for the determination of the Zn , zflnm 7! k) and 231nm 7/ k) coefficients. Hence these coefficients nn may be evaluated by open- and short-circuit measurements. The determination of 22km 7’ k) involves the small difference terms of voltage hence cannot be measured with sufficient accuracy. The determination of an (n 7! k) requires all but one winding be Opened which is an impossible situation. However, these terms need not be measured in view of the conditions established in this section, i. e. n n n znk — an ’3" Znn ' 2 III. GRAPH EQUATIONS 3. 1. Circuit Matrix Formulation In a later portion of this thesis, the following definition and theorems are useful in establishing the requirements sufficient for formulating the system equation with the ampere-turn variable appear— ing explicitly. If a graph G' is not connected and consists of parts p1, p2 . Pk: then at least one vertex of Pm may be identified with a vertex of Pn such that the new graph G is separable into parts p1, p2, . . . . pk and is connected. The preceding identification of vertices does not alter .the segregate or circuit equations associated with the graph G'. Definition 3. 1. 1. An f-circuit is a fundamental circuit that is defined by a chord and its unique path through the tree. 12 Definition 3.1. 2. A subtree Tj-i’ of a tree Tj, is a connected subgraph of Tj, where j = l, 2, 3, Definition 3. l. 3. The complement, Cj-l’ of a subtree Tj-l’ contains all of the elements of graph G in the complement of Tj having both vertices in Tj-l' Theorem 3.1.1. If there exists a subtree Tj-l’ of a tree Tj, in the system graph G such that the complement of Tj-l is nonempty, then the fundamental circuit matrix for the graph G can be written in the form 8 11 0 QL 0 8y = (3.1.1) 821 822 0 Cu; Proof: Let eC be any element of Cj-i’ 6C is a chord. There exists one and only one path in Tj spanning ec. Both vertices of eC are 11 v 12 in Tj—i by definition 3.1. 3. Tj_1 is in T T34 spanning eC . j' Hence there: is a path in The elements of 6 21 and 822 correspond to branches of the tree Tj and the elements of 811 correspond to branches of Tj-v and the first row corresponds to the f-circuits defined by Cj_,. Theorem 3. l. 2. If there exists a set of subtrees, Tj-l: Tj-z: . , To of a graph G with Tj—i in T-, Tj_2 in Tj_1, .a,Toin Tl, then the f- circuit equations for the graph G can be written in the form of 3. 1. 2. 811 6,, o o.........,.o 821 622 8 23 0 631 632 633 634 8-3-1, j-2 0 - 3mm) 8j_1’ 1 ......... ' ..... Bj-1’j_1 6j-l,j — 6a 0 . . . . . 0 i FalI‘GiI‘t') 1 o @L . (Una): m 0 Cch-Jltl o . . . .. o‘M._ fifcflt.) _ ‘UZM olfj-l(t) (wt) (3. 2. 2) where Mjlt) correspond to the variables associated with the branches of Tj and alrej‘t) correspond to the variables associated with the chords Cjoij(j=l,2,3. . . .). Proof: Partitioning the matrix j-l times and applying theorem 3. 2.1 j—l times, the theorem is proven. IV. PROPERTIES OF SYSTEMS CONTAINING MULTIWINDING INDUCTORS 4. 1. Terminal Equations of a System Containing Multi—terminal Inductors . In a preceding section the terminal equations and the transform- ation equations for the currents and voltages are given for a k winding inductor. In general a system may consist of many inductors each having different numbers of windings. In anticipation of future develop- ment, it is desirable to partition these equations into four groups. The general form of the terminal equations for a system contain- ing any arbitrary set of k-winding inductors can be arranged in the following general form: 21TH“) (€11 0 0 0 \mtlh) firm) 0 8.. o o 3,,m Wei“) 0 0 £33 0 $’c1(t) olfczft) 0 0 0 844 $3.“) b 6m“ “”712 ”77713 97]“ 3t1It) 577721 97722 97723 97723 \N’tzul + .5 (4. 1. 1) W31 07732 W33 W34 ‘Q'mltl W41 977742 W43 W44 ‘g'czm It is shown in the following that under certain topologies the sub- scripts t and c designate tree and chord variables respectively. The detailed form of the submatrices in equation 4. l. 1 are best exemplified by an example. 14 Example 4. 1. 1. Let the system graph contain two two-winding, two three~winding, and two four-winding mutually-coupled components as shown in Figure 4. 2. l. 1 7 2 8 3 9 1 4 10 14 V two-winding three-winding 5 11 15 17 6 12 16 18 V four-winding Figure 4.1.1 The resistance coefficient matrix is diagonal, hence for the purpose of this illustration, only the inductance coefficient matrix is considered. Let it be written as in 4.1. 2. Comparing 4. 1. 1 and 4. l. 2, the entries in the voltage and current matrices are ordered as follows: 1‘. Each vector CU‘HU) and [gzfiti] contains one element of each of the mutually coupled components with the elements ordered such that the first entries are the variables associated with the components with the least number of windings, the next entries are the variables associated with the components with the next to the least number of windings, etc. The vector Mi“) consists of the variables associated with the components with the least number of windings, except when the system contains only components with the same number of windings, then at least one but not all of the variables are placed in Wm“). 8.13 J I fl 3.3 32 o o _ _ 3.: o _ o 0 M52 o as . . o 32 o o _ o o _ 2 o o o "~32 o o quvva O 0 «av _ _ O ~Lm2 O mmmz O — O O O I Auvoxr n 2 o _ o o _ o _ 0 3mg 0 o «.3 .m o o 0 mm _ _ o o o O 3*: o . O O Aim.) III.. E o o o a: .I I II _ o _ o _ 2 o o IIIII _II I I I I o o o o n32. o 34> SJ 0 _ _ IIIIIIIIIII _ o o o :2 o _ o o 0 _-§ 0 _ o I I II. IIIIIIIIIII 0 3n». 5 I I I . I I I 0 J2 _ O o O O ”N 1 111111 I. IIIIII _ o o o o o o _ 2 o 3.3 3%: 2:2 o _ _ IIIIIIIIIIIII _ o o o o o S . o o _o :5 III. lllllllll E 31> E o a 2m III 3v m o mEE O _ _ H2. 0 :2 o o O _~_3 III IIII A 3 o _o o _ .2: . _ E o o o d a so «.22 o o o _ _ o 2 o 252 o o _ a a. o o 32> I a a 3...: e n _ o o _ 222 o 2.: _ o 2 o o o . o .3 E o o 31.; E O O _ O O _ :m 0 O Rama: 0 O SVSM o o #32 o _ _ o I :2 o 222 o o _ o :2 o o 0 five; o 2 u u 2 _ O . _ O ”V a o o o n22_o _ o o o 0 332 o _ o o S: o o 0 A32.) II I IIIIII o 2 A S | I l _| IIIII n I0 o o o o 222— o o o 6334/ s a .52 _ IIIIIIIII _ o o o 52 o o o ._ o _ IIIII . IIIII o o a... An : O w-~2 ea lllll w w o 2:2 o _ _ o 22 o o o .22 ||||| IIII “HVOAH o _ O O _ o hmfia m _ 2 o O o . o o «22 o _ _ 2 o 52 o o _ :2 o o 31; 31; _ o o _ o o 0 So _ o 2 o o o o . o o o 2.2. o o _ o :2 o _ o o 222 3.; a A: a o o o o .32 _ o o o o 0 n32 " o o o o fives, fi 3: _ o _ o o _ o o 2.2 o . o . o o . o 3 .5 . o o o _ o 2. o o o v _ E _ O O _ O O O wwz _ o o o o _ o o o 31> _ o o o :2 - a: 16 The vector Whit) consists of the remainder of the variables associated with the mutually coupled components. These variables are ordered such that one variable of each component is placed in the first group, then one additional variable from each component in the second group, etc. , until all the variables have been included. There are several important properties of the terminal equations. Property 4. l. 1. The terminal equations form a symmetric matrix 1f Mnj = Mjn' Property 4.1. 2. M43 = all)“ : 0, when the variables are ordered as shown in 4.1. 2. Property 4. 1. 3.. If the system contains only two-winding inductors then Mag: 6“] 33, °m 13, Om 21, CW)“, (77“ 14' are all zero matrices and all nonzero submatrices are diagonal. Property 4. 1. 4. If the system contains no two—winding components then in 4.1.1 0m“: OYYL_.,4= Oand “Tn“, 6m... qr)... are diagonal matrices. 4. 2. The Current Transformations for Systems Containing Multiterminal Inductors. The detailed form of the current transformation for an arbitrary collection of components can not be shown explicitly. However, the pro- cedure can be established by an example. It is hoped the extension of this procedure to other systems is evident. Referring to 2. 3.1 and retaining the ordering used in example 4. 1. 1, the current transformation equations are as shown in 4. 2. 1. For convenience, in general the current transformation for an arbitrary system can be in the partitioned matrix form. 33m - “Y1 .. o o o 4m) 49:2“) 0 ”In 0 0 41.20:) 402(k) on ,3 ma Ou, o &cl(t) 32.02) ‘11.. “II... o '11. sent) (4.2.2) (I . . I. I4 I 79.. .Q. III. \a.- 3 ‘fl u.". d I ‘2 “IE : .N i :3... . . i o O _ 35 o _ o o _ 2:: :5: o _ o o . o m: o _ 35 o 33 _ o o _ o _ o m: o 3 .5 o o o o G o G I I l I I O H _ O _ O 0 #«JH O _ O O O IIIIIII h I o . o o _ o o 35 3% E? I I I I . I o o o o . o _ II I o 85 _ o A v o o o . IIIIIIII o o m ”can 3? _ H o I I | _ o as w o _ _ o _ I I I I o o m III I o o o _ o _ o o o o _ IIIIIIIII .1 3% N IIIIIIII _ I H o o o _ o o o I I I 3:: I I I I _ I o o o S: . o o _ _ IIIII o o _ o 3 NH 2 o o _ o IIIIIIII o o o a. 3 a o o _ o _ 85. o I _ IIIII o o 2.2 3 H A 2 o o _ o _ o o o _ IIIIIII I vs“ 3 a o 0 III 0 o o _ . o 3.5.. o _ o o o o Aim: 0 _ O _ O O _ O O . o 0 Gym: 0 O _ O O 9: O O . v o o o _ G: o o _ o o o G A”; awn _ O _ O _ O O AuvP—H 3va o _ o o _ o o o o o o .1. five: ... IIIIIII . I o _ o o o _ o o o Ewe E J I I I I I o o m o o o o o _ _ IIIIII 0 «EL o o 3%.: 3: o _ o _ IIIIIII _ o o o a. w o _ o _ o I II I o o 2 o o o . o o o o _ IIIIIIII CV aw So: 0 o u o o _ o o o _ 357 o I I II I o o o o o E: o o o u o o _ o o o o o n o 25- o o o 3.2% o . 33 o o o o o o _ o o o A : o o o _ _ o o o o o 3:? a an at: O _ O _ O O O _ O O O r . o _ o . o _ o o 30: I I o o o _ . o o o o _ o 2:. o o a. _ o o o _ 0 CV oh _ o o o o a o o o s? o _ o E. o . o o H. o o o :c- a .. Zena 18 There are several important properties of the current transform- ation matrix to be used. Property 4~2.1. If the system contains only two-winding inductors M14: and “I23 are both zero and ‘YIB and (T124 have. the same entries as mm and mzz: respectively. Property 4. 2., Z" If the system contains at least one two-winding inductor in addition to other multiwinding inductors, thencnn = 00 As a solution to 4. 2. Z, we have sum =- WI :11 w 0 _ $(:IItI : mm m1: 31:2“) mm :11 Property 4., 2.. 3. - “11;; m on; W124 ’ l ,22. o o OIL 0 0 “LL $31M 3.4%) ‘g' 21“) \I 22m (4.2»3 If the system contains only two-winding components then mm a“: and m“ on; are negative unit matrices and ”In“; —1 and II 23 I“ 22 are zero. 4.3. Components . The Voltage Transformations for Systems Containing Multiterminal The coefficient matrix for voltage transformations is chosen as the inverse transpose of the coefficient matrix in current transformation equafions. ovflml 16TH: 0 “III; “I: I“; ”In GIL-21 OUJ ~14 0m qt on 0 CU) -17 ll -l 22 _ 7 M1“) OVtZIt) WCI“) Guam (4. 3.1) 4. inductor are established. , Instead of pre and post multiplying the original terminal equations by the voltage and current transformation matrices reSpectively, which may be tedious, it is convenient to regard the terminal equations written in terms of the ampere turns variables (the I‘ new variables) as the given terminal equations. “U‘tlItII OU~t2(tI Wei“) _ Weak! 4. The Transformed Terminal Equations“ In Section 2. 3 the transformed terminal equations for a k—winding O ‘71” “61122 W123 0 CU. 0 19 ”7114 ”Ila O ‘U, NEH-I n t2 It) M21“) _ M2203; (403.32; The coefficients of these equations can be determined directly in the laboratory by the methods discussed in Section 2. 3. It is next shown that it is possible to formulate the equations for the system from the component terminal equations given in terms of the ampere-turns variables . I Let the ampere-turns form of the terminal equation for the com- ponents be arranged in the following order. maul-1 _6U22ItI_J GU‘IrtlzItI OUEIItI rn 11 n 21 + n 31 n 41 I'n RM 11 R21 n 31 n l 11 £12 n 22 :6; n 5642 n 12 R'ZZ n 32 n 2 n 13 56?; n 5633 n 43 n ’le n R23 n 33 R23 iii n 2A: n 34 n 44 n 7 I814 11 ‘24 n 34 n 44 clam— 3'22“) 3'31“) 22(tI_ h- - 31:11“)— $3“) $21“) rClzItL (4.4.1) 20 Again it is to be emphasized that in any practical problem the terminal equations for the high permeability, nonlinear transformers would be given in this form. The detailed form of the coefficient matrices in the above equations are best examplified by an example. For comparison, the ampere-turns terminal equations are established for Example 4.1.1, and given by 4.4. 2, where Rii = Rj + njiRi' The subscripts i and j refer to the variables associated with the elements classified as branches and chords, respectively, when the components are included in a system. There are several important properties of these ampere-turns terminal equations. Property 4. 3. 1. The terminal equations form a symmetric matrix n . n . _’ 1f L.nj _ Ljn, 1.e. Mnj _ M3-n Property 4. 3. 2. £2,“ 43, 5634, and ii; are zero when the variables are ordered as in 4. 3. 2. Property 4. 3. 3. If the system contains at least one two-winding . . . . n n inductor in addition to mult1w1nd1ng inductors, then x 23: 32, I. 23: n and 6632 are zeros. Property 4. 5. 4. If all the inductors in the system contain only , , n n n n two—Windings, then 6623, 32, 12, 21, 65“ ,4, 4,, [923, n n n n n . . £32, £12, 21, £14, and £41 are zero and the remaining sub- matrices are diagonal. Emma axon no @3003 3.4.3 Terms- Ire o o o u o o “ om o em 0 o o " rm 0 o o o o I -3m>- 3% o J o o " o o n 0 mm 0 mm o o u o J o o o o 3%, 3% o o J o _ o o _ o o o o «m o _ o o rm o o o 3%, 3% o o o 9m n o o u o o o o o mm H o o o mm o o 3%, 3% . o o o o Tm o n o o o o o o u o o o o E o 3%, 3% o o o o u o E m o o o o o o u o o o o o 7m 3%, m 3% J o o o u o o “gem o o o o o m J o o o o 0 3w? 3% 0 mm o o . o I o _ o 57m o o o o _ o mm o o o o 3 at, 30% 9m o o o u o o n o o 22m o o o n J o o o o o u 3”? 3% 0 mm o o _ o o n o o o 22m o o _ 0 mm o o o 0 3w... 3% o o .m o n o o u o o o o 23m o n o o «m o o o 3”.» new. .oImIohfibIm-_Iwb--m-wloi....mi _IoIo IOWIOI 0 new 3% mm o o o H o o n J o rm 0 o o “22m o o o o o 3w... 3% 0 mm o o _ o o _ o J o J o o u o 57m o o o o 3w... 3% o o .m o n o o u o o o o 7m o . o o 22m o o o 36%, 3% o o o mm H o o _ o o o o o 9m u o o o am o o 3%.. 3% o o o o _~m o n o o o o o o u o o o o 3m o 3%, 3m; . o o o o n o E u o o o o o o u o o o o o :m I 3.3%, AN .¢.¢v Em. - £2 3: o C- O O . A: «J O ”Mm-.2 _ O :- o _ o _ f A o o _ . i 3% o :2 _ o o. c o .24 I I I O O _ _ O msA C O O O _ O _ NIH o _. w." 3% I I I I o 3.2 u o _ o c o 9.me o b m4 0 I I - o _ o o . o A”; 3H. O O . u I I I. C _ O O we. _ O m-~ C O _ - I _ 0 TIM _ 1H “H“ I I I O O O _ NMHZI I _l I I O O O C O - O C o 3va I I .. O o ._ o __ o - I I I I I O «34" o eofiwim C. of I I I I I _ O :HZ . O O I I I ICI “ O O 33 cue -_-- "o o o _.--.I o 2.4 C O o _ I l I ‘ O O O _ I I 333: o 3:4 u o _ I I I - o o _ o o C, “N cf C O O _ O “ 35 I I I I. I O _ O O 3.5 s as o _ o _ as o .5. - - - - - _ o C - O O _ O _ 0 ha CWH O I” l I O 3va o mgr-H _ O _ - _NIHMJ E o o . 53 I I I I I c c o o . o . ES 0 SJ _ CA 0 3:; D O «3 a O _ C4 O 9: C O O _ o S 1H . 0 film 0 . O E: I I O C O _ D . O 53“.; O _ Elm O -. O O ¢fi4 _ I C H C‘IH O Lam-L O _ waN‘q 32 -. - . - can 0 . o o t4 o o r c o o P Q: - .. I O _ m u S 1H I I. O . O «.5. 0 fl: :1 c o o o_-I.-. l o o 0 “so“ 0 c-Ho S o m: c . I o _ 33 as o o H o o H :3 .. ,. I - - - 0 5mg. 0 sow-H 3:4 0 O «A: _ o _ CJ 0 32. I I I I I I __ O o C O 51H 0 _ O .— O FAA CIH O O I. I I I O AHV w." 0 _ O 3 HM‘IH 3 MIL“ O . NANA I I I I C 0 ~04. O _ I 7-H _ 51H 0 Tina. o o c _ o _ O o o C o o H o I o o o _ mm o n o 0 SS _ 0 3:1H 0 o . CA _ O o :4 o . c o o _ o _ o o __ o 0 so . O E . C O mg‘IH :H J _ O C _ O C C _ O O O _ O o o o n o O O O _ o O n O o O o V. SUFFICIENCY CRITERIA 5. l. The Original and Transformed Circuit Equations Let the system graph be such that the fundamental circuit equa— tions Can be partitioned in the form where the indicated partitioning is the same as in 4. l. 1. 6-.- 2.4 Q Chm :t P 1 MneIt) qrrt ('3) Wu (t) Wtz It) “If.- (t) ‘ ° WCZIt) ‘Ifrcm IqrnhIt)_ (5.1.1) Applying the transformation of variable in 4. 3. 2 to 5. 1. l, the transformed circuit equations become: O11 O12 6I21 6I22. 631 832 841 642 Q 13 onus _ 623 0Q” + 6-3 I623 833 8.3 + a 14 “MIMI °fli II “II, II % 11 m 11 3 24 “In 833 abs + 6340(I23 643 00-13 + 644 I)“ 23 6 14 m) 22 824 Orb 22 634 m 22 844 onzz 613 (“14 23 WneIt) okrrt It) Wu ('5) Wtz (‘5) +6.40%. 62307).“ + 8245732491 0 833 67214 + 834572-24 64391-4 + 6... 922.. (5.1.2) 0 0 Cl“) 0 CIIZ‘zItI‘ % 0 cII‘rCItI o ‘M “III-(t) 24 5. 2. The Original and Transformed Segregate Equations Using the same formulation tree as in 5. 1. 1 the fundamental segregate equations may be partitioned in the form12 {CIA 0 O 0 .3“ .3.” 3,3 .2147 \gneIt; 0 ‘LL. 0 o A“ 3,, .32., £2, gum” 0 0 GIL 0 II.“ 4,, J33 i... III-uh) Lo 0 0 CULIIH .8“ 13.4, A“ Mm“) : 0 I &C1It) minim I‘Q’rcIt) ICI‘I (tL (5.2.1) where the indicated partitioning is the same as in 4.1, 1- Substituting 4. Z. 3 into 5. Z. l and part1tion1ng, the transformed segregate equations become: CU. 0 I‘X‘llmu ”If ngqIHIWII (2)211 can J7 AMMZAIDII; \QIneIt - 0 OIL (3.316711; I 3.42. GYIIéIWIIIl (damn + AzZOIIZ/II II'zz1 \QrtItI 0 0 I3211W112I " .2152 OHM‘RIILIII-il (inch-a3 + JJZOIIZAIOQZ'ZI &t1ItIJ 0 0 (34.1%” J’ 342 OIIMI‘CYIIII (Among ”I a44zorIz4+cuaIcII§z fltzItI «84 11 48112 £13 £14» Q2 Icl'III-II _ A! 21 12:22 £223 £24 ‘9, :2 It) I L A31 “3‘32 A33 134 Q, rCIt) 4’41 J42 £43 £44 CI‘I (Q) (5.2.2.) 5. 3. The Location in the Network of the Inductor Elements- A sufficient condition for the ampere—turns of each inductor to appear explicitly in the system equations is 25 (ialmmi'Aqum 'MI One-1i (331%; + 332%)‘112; (441mm- + 3426YI14I0II :11 (13-4in23 + £4.qu24 ‘MImgzl (5. 3.1) 710‘ Referring to 5. 2. 2, the coefficient matrix on the left is non- singular, hence the variables on the left may be expressed in terms of the variables on the right. Therefore the total ampere turns, variable FF“— of each inductor appear in the equations of the system graph eXplicitly. ’3 The location of the elements of these inductors must be investi- gated in order to determine whether the determinant of 5. 3. 1 is non- singular. EI Definition 5. 3. 1. Let Sj be defined as the set of inductor elements I: in the system. Definition 5. 3. 2. Let S- 1 be defined as a subset of S ,_ = 1.2. J’J Theorem 5. 3.1. If there exists a subtree T1 of a tree T, in the system graph, G, which contains no elements of Sj and if the complement of T1 contains at least one element of each inductor, then the ampere- turns variable for each inductor in Sj will appear explicitly in the system equations . Proof. Let the elements of o\f‘ne(t), otfrtu), Wu“), andotrtzh) be in tree T. Also let WneIt) and MrtIt) be in T1. By Theorem 3.1.1 813, 614, 623, and 62,4 in 5.1.1 contain only zeros. If the same tree is used to formulate the circuit and segregate equations, then 8ij = Jj'i 11 Then .331, J41, [32, and $42 in 5. 3. 1 contain only zeros, hence the determinant 5. 3. 1 is non-singular, as c”(IA-11 and “T1,; are non- singular by definition. The theorem follows. The preceding theorem requires that at least one element of each inductor is in a circuit which contains no other elements of any inductor. 26 If the system contains only two-winding inductors the hypothesis of Theorem 5. 3.1 can be relaxed. Definition 5. 3. 3. The set of elements S is defined as a subset cj—1’ of the elements of Sj_1 contained in the complement of the tree Tj-1 of the system graph G. Theorem 5.3 . 2. If all inductors in the system contain only two- windings and if there exists a set of subtrees T3-” Tj-z- . .° . T0 with F” Tj_1 in Tj, Tj—z in TJ'_1, . . . , To in T1, such that the f-circuits defined I by SCj-i’ scj-z’ . . . Sco can be arranged such that each contains one less branch element of Sj’ variables appear explicitly in the system equations. than the preceding, then the ampere-turns ! . Proof. Referring to 5. 1. 1 and considering the first two rows, by theorem 3.1.2, 813 = 814 = 824 = 0. Hence .431: at“ = $42 = 0. According to property 4. 2. 1, OIL-23 = W114 = 0. Thus, 5. 3.1 is non- singular. VI. EXAMPLE 6. 1. Equations to be Solved The previous sections have been devoted to the development of sufficient criteria. on the location of the elements of k-winding inductors in the system graph, such that the ampere-turns variable of the I T. inductors will appear explicitly in the system equations. The application .’ of these criteria in formulating the equations for a typical magnetic amplifier is demonstrated in the first part of this section. The system equations are then prepared for solution on the digital computer to E obtain the output current of the magnetic amplifier, for various input voltages. Many of the steps involved in the preparation of the equations, such that they are adaptable to computer solution are omitted and left to the reader to fill in. The circuit diagram of the magnetic amplifier to be considered is shown in Figure 6.1.1 and its linear graph in Figure 6.1. 2. R5 R1 MAL —’\MM~—————-I Rs 4‘ § 9 > w > ¢ HI {—1 e. (t) ‘5} C > “-4? 4—, o d...» 1:? + (4: (“I Eq fa. Figure 6.1.1 27 28 g l 3" Ire-011:) III 2 3A VII elm )8 Figure 6.1.2 The ”dots" on Figure 6. 1. 1 signify the winding orientation. Choosing the elements 4, 1, and 2 as chords, the fundamental circuit equations are —-I I. - )- -I III 0 WM 0 o -1 1 o 1 o o v5(t) V6It) +1 0 qutI _0 +1J L‘1 0 0 O -1 O 0 1 v7(t) V8“) V3“) V4“) V1“) _VzItIJ I O I-‘ O I—- O O I--‘ O The voltage transformation used is _1 Fv,(t)l in“ n34 o o _ F vg‘m v4(t) 0 1 O O vil(t) v1(t) o o 1 o v?(t) _VIZIII_J _0 O n21 'nzl J _ VilItId (6.1.1) (6.1.2) 29 Substituting 6. 1. 2 into 6. 1. 1, the transformed circuit equations become: V1 0 -1 1 o 1 o 0 Tv5(t)1 o- o eloitl] 1o ..01o1001o v6(t) -1 0 0 0 “34 ‘n34n21'n21 V7“) Veit) v3(t) V4“) V1“) Jami (6.1.3) The fundamental segregate equations are: F 1 o o o o o o 1 1 to “I ’ 110(t)' o 1 o 0 o o o o 0 1 19m 0 o 1 o o o o o 0‘ 1 i5(t) o o 0 1 0 o o o -1 o i6(t) 0 0 o 0 1 o o 1 o 0 17m = o o 0 o o o 1 o -1 -1 0 18m (6.1.4) L o o o o 0 o o o o 1 J 4 13m i4“) i1“) b ia“) _ The current transformation used is EMT in“ o o o q rifltfi 14m = 1 1 o o if“) (6. 1. 5) 11m 0 o 1 1 1110:) Liz(t)_4 LO 0 o -n12q 113W 30 Substituting 6. 1. 5 into 6. 1. 4, the transformed segregate equations become ' 1 0 o o o o 1 1 1 1 1 F 110m“ 0 1 o o o o o o 0 en” i9(t) o o 1 o o o o o o .n,,2 15m 0 o o 1 o o o o -1..1 16m :0 (6.1.6) o o o o 1 o 1 1 o 0 17m 0 o o o 0 1 -1 -1 -1 -1 18m I...” Lo 0 o o o o 4143 o o -n12 - 13m 1 13m 1%) Light i Removing the first two equations, which contain the voltage source current, the reduced set of transformed segregate equations may be written in the following form: "15m " " o o n12 ‘ . ’— ,n “1 i7“) = ‘1 0 rJL121134 iii“) (6. 1. 7) is“) 1 1 (1'n12n34) _, 13th ,n _13 md ,_, 0 0 'n12n34 _J where the two variables, 12m = n34i3(t) + 140:), film = n21i2(t) + 11m, are the ampere-turns variables, one associated with each inductor. The leakage inductance form of the inductor component terminal equations along with the terminal equations for the other components in the system are {vsm ‘ "R5 0 o o o o o o ‘ Visa)‘ v6(t) 0 R6 0 o o o o 0 16m v7(t) o 0 R7 0 o o o o i7(t) v8(t) _ o o 0 R8 0 o o 0 18m . v31“) _ o o o o (R4+n§3R3) R4 0 0 12m vat) o o o 0 R4 R4 0 0 12m Vila) o o o o o 0 R1 R1 1210;) wife). DO 0 o o o o R.(R.+n.2.,R.1..i’}(t)i (6.1-8) 31 F o o o 0 o o o o o o o o o o o o o o o o o o o o o o o o o o o o + 0 0 0 0 (L43+n§3L34) L43 0 0 o o o o L,3 M44 0 o o o o o 0 0 M11 L12 _ o 0' 0 o o 0 L12 (le+nf2L21)_ d is“) 15(t)1 is“) i7it) i?(t) 12m 1%) 11310.. Substituting the transformed circuit and segregate equations into 6.1.8, the system equations result: rii‘m' 1%) L i3“), ' 13W 1%) relo(t)- —R.+ R7+Rs R8 - nlzna,(R,+R7)+R8( 1 - n12n34fl exo(t) = R8 R1+R6+R3 R1+R6+R3(1-n,2n34) _eg (t)J L O O -n12(RZ+R3+R5) _ F- M44 0 ' 111an L34— 0 -3411 1412 Egg- L-n34(M44-L43) n21(Mu"I-112) -n12(L34+L21) _ ,n _12 (t)_ ’(6.1.9) In section 2. 3, it was indicated that the leakage inductance co- efficients are small in magnetic amplifier circuits, and will hereafter be assumed zero. It is also assumed that the two inductors are identical. _ _ _ _ _ 2 - Thus, “21 ‘- n34» 1112“ D43: n21n43 - n12n34 - 1: n21n34 - “12: and R1 - R4, R2 = R3. ~ Solving for the highest order derivatives reduces (6. 1. 9) to a system of two differential and one algebraic equation. " d D'M44'é't" if“) _ 13' K5 em“) ,n LD'Mll-a-t 11(t) D. K6 69(t) _ ,n _ (R0K1+K1K3+R3K2) (R0R3+K2K4+R8K3) 14 (t) ,n L(RoKai‘KzRai'KlKfl (RoKfiKzKfiRaKfl 11 (t) .n .n 412169“) + K214 (t) “K311(t) inm ’- R0 + Kz + K3 (6.1.10) (6.1.10a) 32 Where: Kl : R1 '1' R7 + R8 K4 : R1 + R6 '1' R8 K: = n3. (R1+ R._ K5 = «In (R1+ R?) K. n3. (R1 + R.) K. = n21 (R1+ R.) D' 3 R0 '5‘ 1131 (R1 + R6 '1" R1 '1' R7) R0 : 2R2 + R5 If elements R6 and R7 are equal, then 6. 1.10 and 6. 1.10a become ram (1 ,n ; (R0+2K2)M44—? 14 (t) : Ro‘l’Kz " K5 810(t) _: .n i (Ro+ZKZ)M11E{11(t-)_ R0+Kz K5 69“) .n i - T1 T2 1:1“) (6.1.11) E T2 T1 il (t) r; n n e (t) + K (in(t) in(t) E . : " 21 9 2 4 - ' 1 1 12(t) R0 + mg (6. .11a) Where Tl : ROKI + KIKZ + RBKZ Tz RORB + 1(le + R,,1 Figure 7. 4. 39 m .N. onswflh 30 >0 ” op >moo . u up >mo . u up >mo . u up 10H ? r. (WI) (1)81 40 The steady-state solution of i8(t) obtained as a computer solution in the example in this thesis is given in Figure 7. 5. These solutions were obtained using the curve 'A' in Figure 7. 1 as the value of the co- efficient M. A typical hysteresis loop for the type of material used in magnetic amplifier cores is shown in Figure 7. 6. A more accurate representation of this curve is realized when the width of the loop is not neglected. T B / 14—» . J Figure 7. 6 An investigation was made concerning the possibility of incorporat- ing this type of non-linear characteristic in a computer solution. One of the problems encountered is the specification of the curve the flux should follow at the point where the slope of the hysteresis loop changes abruptly. The problem is further complicated by the fact that during the transient period, the operation is on a minor hysteresis loop. This requires that as the flux follow curve 1 from "a” to "b, " and back to ”a" over curve 2 in Figure 7. 6. However, the point "b" can not be pre- determined by the programmer as it may lie anywhere on the c-d portion of the hysteresis loop. A steady- state solution could possibly be 41 realized for the magnetic amplifier with a hysteresis loop of finite width, if the minor hysteresis loop on which the magnetic amplifier is operating is known. In many of the qualitative analyses only the steady-state solu— tion is considered. Since the above seem insurmountable, the core characteristics used in this thesis is shown in Figure 7.1. The size of the increment of time required is a function of the slope of the M coefficient curve. In particular of the M coefficient curve is discontinuous, then the solutions of the equations developed in this thesis do not exist. The "computer solutions" obtained when the M coefficient curve B is used differs greatly for different values of increment. The size of the increment that could be used in this program is limited by the length of time required for a solution. This limitation is due to two factors: (1) the program is written in interpretive language which requires more time per operation than regular machine language, (2) the time re- quired for the solution to reach a steady- state condition is long for the solution of these equations (Figure 7. 7). Also the possibility of computer failure increases greatly with time especially after several hours of continuous operation. This problem was alleviated by writing the program such that it could be restarted at specific intervals. However, even then the time is a very serious problem. As a first correction to these difficulties, it is suggested that a variable increment based on the rate of change of the ampere—turns variable be incorporated into this program. If this still does not alle- viate the difficulties the entire program should be written in regular machine language. Faster computers are becoming available which will help to alleviate this difficulty. Figure 7. 8 shows the effect of the dc voltage of the control winding on the ampere-turns variables of each inductor. Note that the axis of OOH: ow: (9w) (1)”? .oo . on: 43 m .e where All“ fl by ‘n‘ N 1 fl / _uConufiu op... 3% 3% OH: OH (em) 44 e .e 23E doom... .0 :m >o~ .Gyomun .o.m >mH 83L _ _ Ill sell! ON: 3- INH' NH ON (1)”! (em) 45 i111“) is shifted in the negative direction where as the axis of i2(t) is shifted in the positive direction. Neither inductor however, is driven into saturation. The effect of the load resistance on the steady- state linear solu- tion resulting when the inductors are not driven into saturation is shown in Figure 7. 9. It is hoped that the techniques investigated in this thesis will pro- vide a basis for future development. APPENDIX A The Linear Interpolator Routine This routine is designed to interpolate a curve given by a table which consists of a sequential set of function values, f(xi), for arguments xi. Let x0 be the smallest argument, xp be the largest argument and Xa any arbitrary argument. This routine is written such that if xa< x0, f(xa)= f(xo) or if xa > xp, f(xa) = f(xp). The number of points and the spacing between points is left to the programmer. The mathematical formulation is: [f(xn) - f(Xn - 1)] (Kn - Xa) f(xa) = f(xn) - Xn _ Xn _ 1 The following routine is written in A-l (floating point) and requires the following preset parameters: SK — location of xa SN - location of f(xi) SJ — location of xi In addition the addresses of the following orders must be set: 1. The right hand order of 4 to p-l +SN . The right hand order of 5 to number of points . The right hand order of 20 to the location of the next routine . the left hand order of 18 to (p-l) +SN Uer-WN . The right hand order of 1 to the number of times through the routine The individual orders of this program are included here for the benefit of the reader who is not familiar with computer coding. 46 10. ll. 12. 13. 50 26 OK 85 00 83 8K 05 80 83 1K 15 00 83 13 15 10 BS 15 00 86 88 15 10 87 85 15 80 88 SJ 195K 16L 195K 18L pF lSJ 198K 8L 6L SJ 21L lSJ 195K 21L 21L lSN SN 21L 21L lSN 21L 21L 47 Transfer to A-1 Loop 'q' times Place x0 in the accumulator Subtract xa if x0 > xa, transfer to 16 waste Place Xa in the accumulator Subtract x.p if xa > xp, transfer to 18 Loop p times, if necessary Place xn in the accumulator form xn - xa if xn > xa, transfer to 8 if xn < xa, transfer to 6 Place xn in the accumulator form xn - Xn—l Store xn - xn.l in 21 Place xn in the accumulator form xn - xa Divide x1.1 - xa by xn - xn_l, let this be 'm' Store ‘m' in 21 Place f(xn) in the accumulator form f(xn) - f(xn_1) multiply by 'm' store m[f(xn) - f(xn_l)] in 21 Place f(xn) in the accumulator subtract m[f(xn) - f(xn_1)] store f(xn) - m[f(Xn) - f(xn_1)] 15. 18. 19. 20. 21. 8K 83 85 88 03 BJ 1F 19L 48 transfer to 19 Place f(xo) in the accumulator (from 3) store f(xo) in 21 Transfer to 19 Place f(xp) in the accumulator (from 5) Store f(xp) in 21 Place f(xa) in the accumulator (from 15 or 17) Store f(xa) in SK loop to 2 'q' times Transfer to 'b' Temporary storage APPENDIX B DATA TAPE Section 1 a. Three Sexadecimal Characters to specify: 1. The number of points on the curves showing R6 and R7 as a function of i6 and i7. 2. The number of points on the curves showing Mn and M44 as a function of the ampere turns. b. Three Sexadecimal Characters to set the address of the transfer order at the end of: 1. Input routine. . Coefficient routine (Section 1). . Coefficient routine (Section 2). . Scaled-Derivative routine. . Sine-Function routine (TA-l). . Address routine. . Routine for calculating R6 and R7. muomeww . Routine for calculating i8(t). 9. Output routine. 10. Routine for calculating i?(t). c. The points of the nonlinear elements, with each f(xi) followed by its argument xi in sequence starting with the smallest argument, written in floating decimal form (sign followed by nine significant figures followed by a sign plus two figures, i. e. '+ 1 + 00) for: l. The nonlinear elements R6 and R1. 2. The nonlinear elements M11 and M44. 49 50 Section 2 ' In floating decimal form. R1 R2 R5 R8 D21 1 - the turns ratio to - angular velocity Section 3 In floating decimal form. h - increment e9 (t) em“) t .n . . . 11 (t) - the initial value if“) — the initial value i?(t) - Not needed in the problem solved in this thesis, hence 0 R6 R7 is used. If considered nonlinear elements place 0 in each position, if constant place the desired value in each position. Three Sexadecimal characters, read in by the Main routine, to specify: 1. $09k) Number of loops with memory dumps in between. Number of iterations, with no output of memory. . Number of loops through FA-Z between outputs. . Number of times through the entire calculation, with a change of data each time. , The product of 1 and 2 above is the total number of iterations. 10. 11. 12. BIBLIOGRAPHY "A Magnetic Amplifier for Radio Telephony, " E. F. W. Alexanderson and S. P. Nixdorff, Proc. IRE, vol. 4, Apr. 1916, p. 101—29. . Magnetic Amplifiers, H. F. Storm, Wiley and Sons, 1955. . Magnetic Amplifier Circuits, W. A. Geyger, McGraw Hill, 1954. . "Series Connected Saturable Reactor with Control Source of Compara- "On the Mechanics of Magnetic Amplifier Operation, " R. A. Ramey, Transactions AIEE, 70, Pr. 2, 1951. tively Low Impedance, " H. F. Storm, Transaction AIEE, 69, Pr. 2, 1 1950. . E‘ "A Theoretical and Experimental Study of the Series Connected Magnetic Amplifier, " H. M. Gale and P. D. Atkinson, Proc. IEE, 96, Pr. 1, 1949. . Magnetic Amplifiers and Saturable Reactors, George News Ltd. , London, 1958 . "A. C. Controlled Transducers, " Miles and Law, London Proc. IEE, Vol. 103, 1957. ”Some ASpects of Half Wave Magnetic Amplifiers, " G. M. Ettinger, Proc. IEE, Vol. 105, 1958. Class notes for E. E. 491, Electrical Engineering Dept. , M. S.U. (unpublished). The Linear Graph in System Analysis, W. A. Blackwell, Ph. D. Thesis, M. S. U., 1958. A Foundation for Electric Network Theory, M. B. Reed, Prentice- Hall, Inc., (January, 1961). 51 “wwfit: ‘ w llHllliLlIHLlIEHIIWI H "“1101!" NH IHIINH 93 18 0