TIME DOMAIN ANALYSE OF NETWQRKS CONTAINING A UNIEgQRM TRANgMiSSIQN' LINE TIN!“ I09 I’Iu Dogs-co of DH. D. MICHIGAN STATE UNIVERSITY Harlow M. Judson 1964 THESIS This is to certify that the thesis entitled TIME DOMAIN ANALYSIS OF NETWORKS CONTAINING A UNIFORM TRANSMISSION LINE presented by Harlow M. Judson has been accepted towards fulfillment of the requirements for Ph.D. degree in E‘E, fizIQ/fifl Major professor Date April 15, 1964 0-169 ,_ LIBRARY Michigan State University 4 _.__ _ _4 ABSTRACT TIME DOMAIN ANALNSIS OF NETWORKS CONTAINING A UNIFORM TRANSMISSION LINE by Harlow M5 Judson In recent years, there has been considerable interest in the formu- lation of derivative-explicit equations as the mathematical model of lumped physical systems. This thesis deals with the fbrmulation and solution of derivative-explicit equations for connected networks con- taining a uniform transmission line. The incorporation of a transmission line, described by partial differential equations, into a set of ordinary differential equations associated with the remainder of the network is accomplished by utiliza- tion of the Laplace transform in a manner not dependent on the linearity of other components in the network. The basic steps in the preliminary analysis are based on the concepts of oriented linear graphs. A sufficient condition for the applicability of the analysis is that the transmission line Operate in the conventional two-port manner, that is, in accordance with the simultaneous equations 9‘].-- - éi ax " R; L at iii-3: (IV-Cg!- ax ' at Harlow M. Judson and that all R, C, and L elements be positive and finite. The solution of the derivative-explicit equations is shown to be sufficient for the solution of the network. The complete general formulation is carried out for linear time invariant networks containing a distortionless transmission line, and several special cases are considered. The derivative-explicit equations are in the form.of series of terms representing the multiple reflections and time delays associated with the transmission line. It is found that the solutions must be obtained in a step-by-step process due to the time delayed terms. The general formulation, several special cases, and the general solution process are illustrated by example. TIME DOMAIN.ANAL!SIS OF NETWORKS CONTAINING A UNIFORM TRANSMISSION LINE by Harlome. Judson A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering 196A ACKNOWLEDGEMENT The author wishes to express his indebtedness to committee chair- man, Dr. R. J. Reid for many valuable suggestions and patient encourage- ment; to Drs. R. C. DUbes, L. J. Giacoletto, and H. E. Koenig for their many helpful conversations; and to department chairman, Dr. L. W. Von Tersch for his unfailing support while this thesis was being written. ****W**** -11- . ‘Tallvr‘l , CHAPTER I. II- III- TABLE OF CONTENTS Page mmwmmmmm....................11 LIST OF FIGURES. . . . . . . . . . . . . . . . . . . . . iv LIST OF APPENDICES . . . . . . . . . . . . . . . . . . . v IMOWCTION O O C O O O O O O O O O O O 0 O O O O 0 O O 1 IPRELIMINARY ANALYSIS . . . . . . . . . . . . . . . . . . h THE TRANSMISSION LINE ANALISIS . . . . . . . . . . . . . 21 2.0 Introduction 2.1 Transformation and solution of the transmission line equations 2 2 Frequency domain solutions for the fOur cases 2.3 The two general terms and their inverse trans- forms A General time domain solutions for distortion- less lines THE DERIVATIVE-EXPLICIT EQUATIONS. . . . . . . . . . . . 3h 3.0 Introduction 3.1 The derivative-explicit equations for case (1) 3.2 The general expressions for VT and iT in cases (2). (3). and (h) 1 2 3.3 Some special cases EXAMPLES OF THE ANALYSIS . . . . . . . . . . . . . . . . 61 h.0 Introduction h.1 An example of the analytical procedure h.2 The formulation of a reduced set of equations h.3 An example of the alternate formulation CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . 86 5.0 Discussion of results 5.1 Additional problems LIST OF RMENCE O I O 0 I O O O O O O O O O 0 O 0 O 0 91 7111- LIST OF FIGURES Figure Page 1.0 A general network containing a transmission line. . . . . . 5 1.1 Linear graph representation of a four terminal component. . 6 1.2 Two-port graph representation of a transmission line. . . . 6 1.3 An example network for which the two-port transmission line representation is not valid. . . . . . . . . . . . . 7 l.h Graph of the example network using the two-port trans- mission line representation . . . . . . . . . . . . . . . 7 2.2.0 General representation of case (1). . . . . . . . . . . . . 2h 2.2.1 General representation of case (2). . . . . . . . . . . . . 26 2.2.2 General representation of case (3). . . . . . . . . . . . . 27 2.2.3 General representation of case (A). . . . . . . . . . . . . 27 h.l.0 The example network . . . . . . . . . . . . . . . . . . . . 62 h.l.l Oriented linear graph for the example network . . . . . . . 62 h.l.2' Reduced example network and graph for special cases . . . . 66 h.l.3 Capacitor voltage for t < 6 . . . . . . . . . . . . . . . . 7h h.l.h Inductor current for t < 6. . . . . . . . . . . . . . . . . 7h h.1.5 Capacitor current for t < 6 . . . . . . . . . . . . . . . . 75 h.1.6 Inductor voltage for t < 6. . . . . . . . . . . . . . . . . 75 h.1.7 Resistor current for t < 6. . . . . . . . . . . . . . . . . 76 4.2.0 Example network for illustration of redundant equations . . 78 h.2.l Oriented linear graph for the example network . . . . . . . 79 h.3.0 Example network for illustration of alternative formulation O C I O O O O O O O O O O C O O O O O O O O I 83 -1v- Ann" fl LIST OF APPENDICES Appendix Page A. Some definitions and theorems . . . . . . . . . . . . . . 88 U- AWE-‘01. .' ran- : -V-. INTRODUCTION The time-domain analysis of linear, lumped parameter, electrical networks has been the subject of many authors (1,2,3) and the transient analysis of transmission lines has also been extensively treated (h,5,6). Transmission lines are normally classified as distributed parameter com- D ponents which can be analyzed from the viewpoint of either field theory or circuit theory. Under very few restrictions, derivative-explicit equations can be derived for RIC networks such that solution of the equations is sufficient to complete the solution of the networks (1,3). A network solution im- plies that the current and voltage have been obtained fbr all components of the network. The two primary questions considered in this thesis are: (1) under what conditions can a set of derivative- explicit equations be obtained for RIC networks which con- tain a uniform transmission line? (2) Can the linear transmission line be handled in a manner which is independent of the linearity of the rest of the network? In Chapter I the restrictions on the network are stated and the oriented linear graph representation Of the transmission line is intro- duced. A set of modified derivative-explicit equations is then derived and a discussion on the reduction of the modified equations into the normal derivative-explicit form is given. In Chapter II the four cases which can arise with respect to placement of the transmission line graph elements in the tree are con- sidered. A transmission line problem can be associated with each of the four cases and the laplace transform is used to obtain general solutions for the fOur problems. If the transmission line is distortionless, general time domain solutions can be obtained in terms of assumed arbi- trary driving functions. If the transmission line is not distortionless, the time domain solutions cannot be Obtained for general driving func- tions except in terms of rather complex integrals. The time domain solu- tions obtained fer distortionless lines are in the form of series of time-delayed terms corresponding to the multiple reflections which occur at the transmission line terminals. In Chapter III, the series solutions of Chapter II are combined with the modified derivative-explicit equations of Chapter Iito produce derivative-explicit equations in normal form. Each resulting equation is in the fbrm of a series of terms. Each term contains a delayed unit step as a factor so that the terms become sequentially non-zero. As a consequence solutions must be obtained in a step-by-step manner since in general the equations change whenever a term of the series becomes non-zero. Recursion formulas are given for the calculation of coef- ficients appearing in the final equations. In Chapter IV, the general fermulation and solution processes are illustrated by example. Several special cases are also illustrated while the general solution is being obtained. In Chapter V, the major results of the thesis are reviewed and some topics warranting further investigation are mentioned. I. PRELIMINARY ANALISIS Given a connected network composed of a uniform transmission line of arbitrary length and a finite set of R,L,C, e(t) and h(t) elements such that: (1) The network contains no circuit of e(t) elements. (2) The network contains no cut-set of h(t) elements. (3) The transmission line Operates in accordance with the 4 transmission line Eqs. 1.0 and 1.1. L _ CL .- - - can-i— B - Ri L 8' 1.0 ii— — - - a— ; — Gv C 6' 1.1 (A) Certain matrix inverses exist; this will always be the case for networks with positive, finite, R,L, and C elements. Then the following results will be demonstrated. (1) The transmission line, Operationally described by simultaneous partial differential equations, will be incorporated into a system of ordinary differential equations in a manner not dependent on the linearity of the remainder of the network. (2) If the transmission line is distortionless a system of de- rivative-explicit equations will be derived. The solution of these equations is shown to be sufficient to complete the solution of the net- work. The general problem to be considered is illustrated by Fig. 1.0 where N is the entire network except for the transmission line. NOte teat N may be disconnected without violating the condition that the en- tire network be connected. Figure 1.0 A general network containing a transmission line. One of the possible oriented linear graph representations of a four-terminal component (7) is illustrated in Fig. 1.1. For the Special case in which the four-terminal component is a transmission line Operat- ing in accordance with Eqs. 1.0 and 1.1, the element in Fig. 1.1 labeled may be omitted because i v3, i is identically equal to zero. Figure 3 3 1.2 illustrates the oriented linear graph representation used throughout this thesis for the transmission line. The graph given in Fig. 1.2 will be called the two-port representation. a... J-I‘F' '3... 73'1“ v3,i3 Figure 1.1 Linear graph representation of a four terminal component v(O,t) V0513) i(0,t) J) i(£,t) Figure 1.2 Two-port graph representation of a transmission line Two important points can be illustrated by considering the net- work of Fig. 1.3 and the corresponding linear graph of Fig. 1.h. (l) The network currents and voltages obtained by using the two- port representation are not always valid. (2) Use of the two-port representation, even when it is not valid, does give rise to a problem.which can be analyzed. Examination of Fig. 1.h leads to the immediate conclusion that no current can exist in any of the elements since each element is a cut-set. This result illustrates point (2). Point (1) fbllows from.the fact that a current would actually be present in the voltage driver due to the capacitance between the transmission line conductors. o————— flop X=O X=£ <3 — -——o Figure 1.3 An example network for which the two-port transmission line representation is not valid (9 V(‘:t) (0.t) v (We) 1(0.t) Figure 1.h Graph of the example network using the two-port transmission line representation The following definition and theorem specify one class of net- works N for which the two-port transmission line representation is always valid. Definition 1.0: A graph is disconnected if a prOper subgraph and its complement have no vertex in common. Theorem 1.0: If the network N of Fig. 1.0 is disconnected then the two- port transmission line representation is valid. Proof: If an oriented linear graph is drawn for the entire network and the three element representation of Fig. 1.1 is used fbr the transmission line, then the element v3, 13 is a cut-set; hence i3 5 0, which completes the proof. The following definition and theorem.are given as a.possible aid to recognition of disconnected networks: Definition 1.1: A graph is separable if a.proper subgraph and its com- plement have exactly one vertex in common. Theorem 1.L: The graph G of a network N is disconnected if and only 1 if fer some proper subgraph C1 of G, the vertex matrix of G can be written in the diagonal formgg All 0 A = 102 where the columns of AE associated with A correspond to the elements 11 of G and the columns of A22 correspond to elements in the complement of 1 G1. Proof: Follows directly from definition 1.2. The general analysis in the following paragraphs is carried out 1 See definition A(1). 2 Both the f-circuit and f-cut-set matrices of a graph G can be written in the diagonal form of Eq. 1.2 if G is either separable or disconnected. for a general connected network N. The results are valid for networks in which the two-port representation of the transmission line is valid. The graph elements associated with specific measurements at a transmission line port will be called T elements in this thesis. A subscript 0(1) will be appended to the T designation, when apprOpriate, to indicate a particular port. For a given network of the class under consideration, an appro- priate linear graph representation is obtained and a tree is selected by the following rules1 used in the order given: (1) All e(t) elements are placed in the tree. (2) All possible C elements are placed in the tree. (3) The tree is completed with R and/or T elements, if possible. (A) If necessary, the tree is completed with L elements. Several general statements can be made concerning these rules, with proof of validity readily available (2,3). (1) If the graph has no circuit (cut-set) of voltage (current) drivers, it is always possible to place all voltage (current) drivers in a tree (cotree). (2) Any C element not in the tree is in a circuit of e(t) and C elements. 1 These rules are very similar to the rules used in RIC graph analysis (3)- -10- (3) Any L element not in the cotree is in a cut-set of h(t) and L elements. In the course of selecting a tree by the given rules, fOur possible assignments may be made with the T elements. These assignments will be called the four cases: Case (1): T£(To) is a branch (chord). Case (2): 36th T elements are chords. Case (3): To (T!) is a branch (chord). Case (A): Both T elements are branches. The notational convention for the equations in the following analysis is specified by definitions 1.2 and 1.3. Definition 1.2: The branch (chord) L, G, and C element matrices of a graph G are the diagonal element value matrices L1, G1, C1 (I2, 62, 02) where the subscript 1 (2) indicates the elements to be branches (chords) of a tree (cotree). Definition 1:3: The branch (chord) C, R, T, and L current and voltage matrices of a graph G are the column voltage and current matrices ic , 1 i , i , i (i , i , i , i ) and v , v , v , v (v , v , v , R1 T1 C1 C2 R2 T2 12 Cl R1 r"’1’ L1 C2 B2 T2 Vi?) where the subscript 1 (2) indicates the currents and voltages to be associated with the branches (chords) of a tree (cotree). Three basic sets of equations associated with the graph of an -11- electrical network are the f—cut-set, f—circuit, and element equations. The general forms of these equations for the class of networks under consideration are, respectively: ll 21 B3h BSA 45 55 811+ set; S 31+ Sun 551+ 1.3 1.h -12- I—d - r“i _ "E: _ —v Cl 1 dt Cl 1 G .1 1 Q R v11 11 it 1 = 1‘1 105 i C C 2 d 2 -—-v i G at C2 2 R v 12 2 _ _ _ - 9L.1 dt E2 Equations 1.3 and 1.h are often written in more concise form as [U I Sc] --l- - o 1.6 . 12 I v1 [ET I U] ---- = 0 1.7 When the columns of the f-cut-set and f-circuit equations are in the same order, the following relationship holds (Th. A(1)): SC = -Bi‘ 1.8 where the prime superscript indicates the transpose of the matrix. Applied to the submatrices of SC and BT’ Eq. 1.8 implies: __ _ I Sid - Bji , 1.9 The zero submatrices of SC(BT) in Eq. 1.3 (l.h) are a direct con- sequence of statements (2) and (3) fbllowing the tree selection rules, -13- and Eq. 1.9. The general form of the derivative-explicit equations obtained for RLC graphs (1,3) can be written as v v e(t) %§'e(t) d . EI' i = Ao i + A1 h(t) + A2 d 1'10 d't' h(t) where the coefficient matrices, [A1], are specific algebraic combinations of submatrices from the f-cut-set, f-circuit, and element equations. It is also shown that the solution of Eq. 1.10 when appropriately substitut- ed into the three basic sets Of equations is sufficient1 to complete the solution of the network. One of the major goals of this thesis is the derivation of equa- tions similar in form.and properties to Eq. 1.10 for RIC networks which contain a uniform transmission line of arbitrary length. The equations obtained in the latter case are comewhat more complex than.Eq. 1.10 due to the reflections and time delays associated with the transmission line. The variables in the left hand column matrix of Eq. 1.5 can be in- corporated into a subset of the matrix Eqs. 1.3 and l.h, as indicated in Eq. 1.11. 1 The number of Eqs. necessary and sufficient for the solution of the RLC network is equal to the rank of [ 1. Full rank does not result when capacitors (inductors) in series (parallel) are in the tree (cotree), nor under a variety of other conditions. <3 <3 cfl R: -1h- 25 35 The variables of the right hand column matrix of Eq. 1.5 can similarly be expressed as -B11 0 O -B O O 0 0 0'1 0 0 0 3&3 BM 0 0 0 -U_ 1.11 Ti _. d‘t’vc1 le d —i at 12 112 d . 53h“) d 3.36“) e(t) Vol V T1 substitution of Eqs. 1.11 and 1.12 into Eq. 1.5 yields, after multiplication of coefficient matrices and some application of Eq. 1.9: 1 _ i _ s 0 o 0 O 0-1 T2 PO +8 0 s' 0 7 S23 52h 25 iL2 1 21 2 21 _ _ d s O 0 0 0 O O 0 -—av S33 33% S35 h(t) dt C1 . = t O 0 O B1+1 13h2 Bh3 Huh 0 e(t) 0 t2+3h5LiBh5 3E112 I V 0 0 O O 0 O O -U c 0 O — 1 L. _J l I_. _. V R1 V T1 1 _st _ I31 I I I I I - S22G2S32 O S2102511 S2202512 S2252822 S22G2Sl+2 g__h(t) dt I I I I Gl+S32Ges32 O 0 33202312 $32G2822 S32023h2 d A -—-e(t) 1.13 ' 0 O 0 0 dt 0 BhsLlBss ( ) e t [02323 O O “G2B21 'G2B22 "G2Beb. I _] VC 1 v The matrix equation represented by the second row of submatrices of Eq. 1.13 is '5331T2'S3hiL2—835h(t) = [Gl+S32G2Séal le+S32G28i2e(t)+832028é2vcl+ I S32G28h2le 1.1h When the matrix IGl+S32G28é21 is non-singularl Eq. 1.1% can be solved for v as R1 When all network resistors are positive and finite, the matrix is positive definite (Def. A(2)) and hence non-singular. -16- -l _ _ I vR _ A S331T +S3hiL2-I-S35h(t)+S32G2812e(t) + 1 - 2 _ t ! S32G2S22Vc1+s32G23h2vT1 1.15 where A=[G+S GS'l 1.16 1 32 2 32 and the -1 superscript indicates the matrix inverse. The matrix equation obtained from the fOurth row of submatrices of Eq. 1.13 yields an expression fer iR2. Substitution of Eq. 1.15 into this expression gives -1 ' . iR2 = 0.232311 . S33iT2+83h112+S35h(t) + [S32G2512+C'25i2] e(t) I I I I +[S32G2822+G2822l vCl + I532G2su2+G2Sh21 le 1.17 Substitution of Eq. 1.15 into the matrix equations obtained by expanding the first and third rows of Eq. 1.13 yields a set of modified1 derivative-explicit equations which can be written as l The word modified is used because of the presence of the unspecified quantities v and i in the Eqs. T1 T2 -17- d d vc1 -1 Seiczsii 0 It e(t) 5" 1 {“3 0 s s 9—h(t) + 1*2 15H 55 dt Scs' 'ls GS'-SGS' scs' '13 -s_ —v . 22232A 32222 22222 22232A 311211 C1 + -1 - Séh'séu" S3292322 “BB" 13113 1 112 " -1 -1 ‘ ' ‘ S22G2’5’32A S326251'2'622623312 S22625§2A 935'325 e(t) -1 ' - + sin-351A S32G2812 4313A 11353 a Lh(t)d " . -1 , -1 " ' " 52292352“ S32523112"3226231i2 522°2S§2A S33"“323 VTl I '1 ‘ where the matrix B, which is positive definite for positive, finite, L and C elements, is given by I C1+321C2$21 0 B I: 0 12+Bl+51131i5 . 1’19 Writing Eq. 1.18 with notation similar to that used in Eq. 1.10 yields t v d vCl vcl B e(t) + B ;L, e( l + B T1 1 20 E76 112 Bo 52* 1 h(t) 2 dt h(t) 3 1T ’ - ‘ 2 It can be readily concluded by examination of Eqs. 1.10 and 1.20 -18- that if expressions can be found for vT and iT such that 1 2 le = fl(vCl, iy2, e(t), h(t)) 1.21 1T,2 = r2(vcl,iL2, e(t), h(t)) 1.22 where the functions fl and f2 are linear combinations of the variables in the arguments; then substitution of these expressions into Eq. 1.20 will yield an equation which is in the form of Eq. 1.10% The formal procedure for evaluating v and i in terms of T1 T2 vcl, 1L2, e(t), and h(t) will be: (1) Obtain expressions for 1T1 and VT2 from Eqs. 1.3 and 1.h, respectively. (2) Eliminate resistor currents and voltages from the expressions obtained in the preceding step by use of Eqs. 1.15 and 1.17. (3) Use the resulting expressions as drivers at the appropriate transmission line ports. (4) Solve the resulting transmission line problem for VT and 1T . l 2 (5) Eliminate undesired quantities by substitution. Eqs. which have the form of Eq. 1.10 are said to be in normal form. -19- Three questions require consideration: (1) under what conditions can steps 1 through 5 be taken? (2) If the five steps can be taken and the solution of the resulting equations can be obtained, can the solution of the network be completed? (3) Since the propoSed analysis involves obtaining rather general solutions for the transmission line terminal quantities, can these solutions be utilized in an alternative fermulation procedure to reduce the number of equations which must be solved? Question (1) will be considered in Chapters II and III. Question (2) is readily answered. The quantities vC and 112 are 1 obtained in the solution of the equations. Then iC and VL2 are obtained 1 from Eq. 1.5. Since vT and 1T are assumed to be expressed in terms 1 2 of vC , i , e(t), and h(t), these quantities are known, and the re- 1 . I2 Sistor currents and voltages can be obtained from Eqs. 1.15, 1.17, and (i C’2 711 1.h (1.3), i (v ) can be fOund from.Eq, 1.5. Finally, 1 and v can be obtained from Eqs. 1.3 and 1.h since all chord currents and branch 1.5. Since v ) can be Obtained from the first (last) row of Eq. voltages have been evaluated. Question (3) is somewhat more subtle and may require some addi- tional explanation. Suppose that a C(L) element in the tree (cotree) is in parallel (series) with a T element. under these conditions, does -20- step (A) in the proposed procedure effectively eliminate the equation associated with this C(L) element? Consider the formulation which would result if the T element rather than the C(L) element were placed in the tree (cotree). This would imply that the f-circuit (f-cut-set) con- taining C(L) would consist of T and 0 (T2 and L), and that C(L) would 1 be a chord (branch). Uhder these conditions, the fermulation step associated with Eq. 1.12 would involve derivatives of vT and iT , in additional to the quantities which appear in the right hind coluin matrix of Eq. 1.12. This implies that the resulting equation which is equivalent to Eq. 1.18 would contain derivatives of le, and 1T2. It is concluded that the differential equation for C(L) has been replaced by a differential equation for VT (1 T ); hence the number of equations has I 2 not changed. It should be noted that the actual number of equations which must be solved Simultaneously can be less than the number specified by Eq. 1.20. In general, both the interconnection pattern and the component values can effect the number of equations requiring simultaneous solutions An example network illustrating the reduction of the number of simultaneous equations is considered in Chapter IV. The example also illustrates the important point that the equations in the reduced set are generally more complex than the original derivative-explicit equations. Therefore, reducing the number of equations may not appreciably simplify the network solution. II. IHE TRANSMISSION LINE ANALYSIS 2.0 Introduction In this chapter, general frequency domain solutions are obtained for transmission line problems representing the four cases which arose in Chapter I, and general time domain solutions are obtained for dis- tortionless lines. In section 2.1, the problems are restricted to uni- form transmission lines with zero initial conditions, and the general frequency domain solutions of the transmission line equations are ob- tained. In section 2.2, the frequency domain solutions are obtained for the four cases. These solutions are in the form of series of terms representing the multiple reflections at the transmission line terminals. In section 2.3, the two general terms of the series solutions are dis- cussed, and the inverse transform of the general terms is given. In section 2.h, the general time domain solutions are given, for each of the four cases, for distortionless transmission lines. 2.1 Transformation and solution of the transmission line equations. The voltage and current distributions on the uniform transmission line are assumed to correspond to equations 1.0 and 1.1, and the initial distributions are v(x,0) O < x < .6 2.1.0 II 0 s. i(x,0) O < x < .6 2.1.1 ll 0 V -21- -22- Taking the Laplace transform of 1.0 and 1.1, subject to equations 2.1.0 and 2.1.1, gives §_!i§i§2.- - (R 1 5L) I(x,s) 2.1.2 dx - §_IIEL§2.= - (G + SC) V(x,s) 2-1-3 dx Taking the derivative of Eq. 2.1.2 (2.1.3) with respect to x and substituting Eq. 2.1.3 (2.1.2) into the resulting expression gives 2 d g x s) = (R + sL) (G + sc) V(x,s) 2.1.h dx 1 2 9—-%$§L§l- = (R + sL) (G + sc) I(x,s) 2.1.5 dx . The general solutions of Eqs. 2.1.h and 2.1.5, respectively, can be written as: V(x,s) = AeTx + Be-Tx 2.1.6 I(x,s) = Derx + Fe-Tx 2.1.7 where A, B, D, and F may be functions of s, and r = (R + sL)l/2 (G + sc)l/2 2.1.8 Substitution of Eqs. 2.1.6 and 2.1.7 into Eq. 2.1.2 yields A = -ZOD 2.1.9 B Z F 2.1.10 0 where 1/2 R + sL Z0 -|:m] 2.1.11 Thus Eq. 2.1.7 may be written as I(x,s) = %f- I -AeTx + Be-TXI 2.1.12 0 2.2 Frequency domain solutions for the four cases. In general, A and B of Eqs. 2.1.6 and 2.1.12 can be evaluated if two independent terminal quantities are specified. In the fOur problems considered in this section, the specified terminal quantities are v T2 and 1T , as indicated in Chapter I. 1 In the following equations, the notation used is defined by Eq. 2.2.0. Fi(s) sign» 2.2.0 Case (1) will be considered in some detail, while only the re- sults are given for the remaining cases. Figure 2.2.0 illustrates the general transmission line problem of case (1), in which T! is a branch and T0 is a chord. From Eqs. 2.1.6, 2.1.12, and 2.2.0, we have E(s) = V(O,s) = A + B 2.2.1 T2 -T£] H(s) .—. I(£,S) = %—[ -Ae + Be 2.2.2 0 -2h- This last pair of equations can be solved for A and B, giving E(s) e"?r -ZO H(s) e-Tz A = 1 + 62” 2.2.3 E(s) + Z0 H(s) e-T£ B = '1 + e_2rz 2.2:h + e(t) x = 0 x = z h(t) Figure 2.2.0 General representation of case (1). Using Eqs. 2.2.3 and 2.2.4, the unspecified pair of terminal quantities can be expressed as I '2”)] ———-l———— 2.2.5 v(£,s) = I2E(s) e-Y 2T1 - Z H(s) (1 — e O - 1 + e E s -2T£ -T1 1 I(0,s) = [iii—1 (1 - e )+ 2 11(5) e 1 :57 2.2.6 If Ie-2T£| < l, the fractional factor can be expanded in an in- finite series as 1 8-2” + e-hm _ e-6rz 2Y1 =1 “ 'I' 000 20207 1 + e- -25- Since Re(T) >’0, the series expansion is valid unless I = O. For 1 = 0, Eqs. 2.2.5 and 2.2.6 become, respectively: 13(8) 2.2.8 V(lis) I(O,s) H(s) 2.2.9 This last pair of equations are merely expressions of Kirchhoff's cir- cuit and vertex equations. Substitution of Eq. 2.2.7 into Eqs. 2.2.5 and 2.2.6 gives, after some manipulation to collect terms, V(£,s) = 2E(s) I e-Yg - e-3T£ + e-5Y£ ... l - ZO H(s) + 2ZO H(s) Ie-2T1 - e-AYI + e-6Y£ ...l 2.2.10 I(O;S) = §£§)'- 2%122 [e-2r1 - e-hrl + e-6rl ...] + o o 2H(s) [e-T£ - e-3Yz + e-5Y£ ...] 2.2.11 Equations 2.2.10 and 2.2.11 are the general S-domain solutions for vT and 1T of case (1). The inverse transform.will not be con- 1 2 sidered until comparable results have been given for the remaining cases. Case (2): neither T element is in the tree. Figure 2.2.1 illustrates the general problem and the placement of the drivers. The solutions obtained for the unspecified terminal quantities are given by -26- Eqs. 2.1.12 and 2.1.13. 31(8) 231(5) -2” -1an -6m — -—————- [e , + e + e + ...I I(O,s) — Z + Z - o 0 2E (s) 22 [e’TZ + e‘3r‘ + e'Srziz..] 2.1.12 0 E (s) 2E (s) _ _ _ - I(£,s) = 2 + 2 [e 2Y1 + e AT! + e 6T3 + ...l- O 0 2B (8) --lL-- Ie-Tl + e-3rl + e-Srl + ...] 2.1.13 <9 I Figure 2.2.1 General representation of case (2). Case (3): T0 is a branch and T! is a chord. Figure 2.2.2 illus- trates the general problem and the placement of the drivers. The solu- tions obtained for the unsepcified terminal quantities are given by Eqs. 2.1.1h and 2.1.15. e-3Yt + e-STI V(O,s) = 2E(s) [e'T‘ - ... J + Z0 H(s) - [e-Iflz " e-LFY“ + e-6r£ coo] 2Zo H(s) 2.1.1h -27- - - l - l - I 'I(£,S) = 34-5-1 - '2—3313'2 [8 2r "’ e [IT 'I' e 6r coo] " O O 2 H(s) Ie-Yl - e-3rt + e-Srl ...] 2.1.15 d A ___ v e(t) ll IV. h(t) x = 0 x A ___ A V v Figure 2.2.2 General representation of case (3). Case (4): both T elements are in the tree. Figure 2.2.3 illus- trates the general problem and the placement of the drivers. The solu- tions obtained for the unspecified terminal quantities are given by Eqs. 2.1.16 and 2.1.17. ...] - V(O,S) = 20 Bil-(S) + 22.0 111(8) [e-2Tl + e-I-I-Y'l + e-6Y1 220 Hé(s) [e-r£ + e-3T3 + e-STI + ...l 2.1.16 V(£,s) = 220 Hl(s) [e-Yz + e-3T3 + e-Srl + ...] - Zo Hé(s) 220 Hé(s) [e—2r£ + e-hrfl + e-Grt + ...l 2.1.17 hl(t) x = O x = z h2(t) (D Figure 2.2.3 General representation of case (A). -28- Equations 2.1.10 through 2.1.17 are the general S-domain solu- tions for VT and iT in the four cases. In obtaining these solutions 1 2 by means Of the Laplace transform, the problem has not been limited to linear networks, although the Laplace transform is a linear Operator. The only condition imposed on the network N by this fbrmulation is that the linear combinations of element voltages and currents corresponding tO VT and 1T be Laplace transformable (8). It can be concluded that 2 1 this analysis Of the transmission line is valid, in most cases, for nonlinear electrical networks. 2.3 The two general terms and their inverse transform. Equations 2.1.8 and 2.1.11 can be rewritten as 1 R G1 T — G—51/Qs + L) (s + 5) 2.3.0 d s+% ZO = R0 G 2.3.1 8+6- are given by Eqs. 2.3.2 and 2.3.3. L where R and v 0 d 1 v 2.3.3 d LC The quantities R0 and v correspond to the characteristic resistance d and propagation velocity, respectively, on distortionless transmission lines. -29- By using the notation of Eqs. 2.3.0 and 2.3.1, it is readily apparent that only the following two distinct forms occur in the series solutions of section 2.2: -A ~f(s+p)(s+o) Form (1) Kh F(s) e n - U+)s+) Form (2) Kg F(s) EIE' e An ( p ( 0 where: (l) F(s) is the Laplace transform of a specified function. (2) Kn and K; are constants which in general depend on n. n! (3) A =— n vd -32 - . (“I 0(0) — L (C) if R0 18 a factor of Kn' —§._R_"I 3;... 1rI (5) 9(0) — C (L) if R0 18 a factor of An. The inverse laplace transforms of the two general forms can be calculated by use of the convolution theorem and transform tables (8,0) / ',‘ r‘o LAD. A - -1 -AJ s+ ) S ) t -—£( +O) of KnF(s) e n( p ( +0 = Kn] e 2 p (3A f(t-tl) citl + An n t 1 1 -—(p+o)t t - Kn [ go...) e 2 l ——-l—-l- Il<§ < < n 3 To simplify the writing of the rather considerable number of lengthy equations which appear in the f0110wing pages, several notational simplifications are defined: v(z,t) (i(£,t)) = v£(t) (i£(t)) 3.1.6 V(01t) (1(0,t)) = Vo(t) (10(t)) 3.1.7 CO = - [Su2G2B23A-lS33+Su3] 3.1.8 01 = -sh2[02323A‘ls3zaesfi2+oesfi2] 3.1.9 N(t) = 1T1 - COiT2 - clle 3.1.10 D5 = -333A'ls33 3.1.11 D1 = -[B3h+Sé3A-ls32G28i2] 3.1.12 M(t) = VT.2 -D01T2 - 131le 3.1.13 It should be observed that N(t) and M(t) involve only linear com- binations of the quantities 1L , vC , e(t), and h(t). Substitution of 2 1 -37.. VT (-1T ) for e(t) (h(t)) in Eqs. 2.1.3 and 2.1.1 yields 2 1 ROGOUO ] v =-—-———-[cO 10 +N + Z l- -ClRo oo o 2 [( —-:-—- Di +sz+M) 6U -(Di +Dv£+M) 6U +...]+ 1 ClRo 0 ol 1 1 l l l o 03 1 3 3 3 3 2R0 [(c0 10 +0 v +N2 ) e U - (c 10 +01 v +N ) e U + .. 1 z o l- -ClRo 02 l 2 2 2 0041.3)+ h h h 3.1-.11“ eouo 2 i = [D v +M 1 - [(D 10 +Dl v + -(D 10 +Dl v +M )e U +...]- 00 Ro-Do 1 £0 0 RO-Db o 02 £2 M2)€ 0 Oh lflh h h h 2Ro Ro'Db [(Col 1+Clv£11+N l)elU l -(Coio3+Clv£3+N3) e3U3 + ...] 3.1.15 where the notation of section 2.h has been used for time and exponential functions. Substitution of Eq. 3.1.15 into Eq. 3.1.lh yields, after some algebraic manipulation, the following expression for v3 : Ro(Ro-Do) RoCo O V30 = -_—II__—__ NO + D M.O eOUO + 2(R -D ) o o -__IT——-'[(D5101+Dlv£l+Ml) elUl - (D5103+D1V23+M3) €3U3 + ...] - 2R (R -D ) o D? O [(0010 +Clv£ +N2 ) e2U 2 -(C010 +Clv£ +Nh) thh + ...] - 02 £2 h h 2R C o o . D [(Qoio 2+DH1V £2+M2) 62 2-(Dbioh+nhvflu+mh> €hUh + ...] - 2R2C 00 / D [(Coio +C1Vz +Nl) elUl - (Coio +Clv£ +N3) €3U3 + ...] 3.1.10 1 l 3 3 -38- where D = (l-C1R0)(RO-DO)- RoCoDl 3.1.17 Substitution of Eq. 3.1.16 into Eq. 3.1.15 gives the following expression for 10 : o i = l-C1R° N + D1R° N s U - 00 D o D o o o 2(l-c R ) 1 o . ___—Ir___'[(D0102+D1v£2+M2) €2U2 - (Dbiou+nlvlh+Mh) thh + ...] - 2R D o l . D [(ColO +C1v£ +Né) €2U2 - (CoiO +Clv£ +Nh) thh + ...] - 2 2 h “h 230(1-clR0) D [(0010 +ClV5 +Nl) €1U1 - (C01o +Clv£ +N3) e3U3 + ...] + 1 1 3 3 2D1 ‘3f' [(I%101+Dlv£l+Ml) elUl - (Dblo3+D1v£3+M3) e3U3 + ...] 3.1.18 Equations 3.1.17 and 3.1.18 are valid for all values of t, and can be generalized by replacing t by t-nT. The resulting expressions can be written as -3x .. v - M N + ROCO M e U ... 3n D n D n o n 2(R -D ) O O * *- -_-—75-—- [(Dn+l+Mh+l) e1Un+1 - (Dn+3+Mn+3) e3Un+3 + °°°] - 0 Co [(D Dn+2+ Mn+2) 62Un+2 (D n:h+M n+h) EhUn+h + "'1 - 2R (R -D ) o o o * * D [(Cn+2+Nn+2) e2Un+2 - (Cn+h+Nn+h) thn+h + "°] - 2R 2C oD Co * [(C Cn+1 Nn+1) e1Un+l ’ (Cn+3+Nn+3) e3Un+3 + "' ] 3'1'19 l C R D R i = 1 O M + 1 ° N e U + 031 D n D n o n 2D 1 * * _D— [(Dn+l+Mn+1) e1Un+1 ' (pn+3+Mn+3) e3Un+3 + "'J ' 2(1-0 R ) l o * "“73“" [(Dn+2+Mn+2) e2Un+2 ' (Dn:h *Mn +u) th n+h + °"] ' 2R D o 1 * D [(Cn+2+Nh+2)€ 2 Un+2 (Cn:h% +h) thn +h + ...] - -2R (l-C R ) o 1 o * * D [(Cn+1+Nn+1) elUn+l - (Cn+3+Nh+3) e311n+3 + ...] 3.1.20 where * Dn- = D010n+DlV£n 3.1.21 * cn_ = coio +C1V£ 3.1.22 d * an Cn where -ho- * By using Eqs. 3.1.19 and 3.1.20 in Eqs. 3.1.21 and 3.1.22, Dn are evaluated as 2 D R J 1 o DMn+ D Nn eoUn+ 2D R 1 o * * D ( n+l+Mn+l) €1Un+l - (Dn+3+Mn+3) e3Un+3 + "'1 - 2D1Ro 2 [(c* n+2+ Nn+2) €lUn+2 (Cn:4+Nn +h) thn +h + "'J ' 2J [(D* n+2+ Mn+2) €2U n+2 (D* n+l++ Mn+h) thn+h + "’ 1 ' 2RO J C R K ._2. .Jl. [DD Mn + D :l eOUn + 2K * [(D Dn+1 +Mn+l) e1Un+1 ' (Dh+3+Mn+3) e3Un+3 + "'1 ' 2R K o *- -73—- [(Cn+2+l\ln+2)e 2 Un+2 (C n+h+Nn +h) thn+h + "°] - 2C 0 * . T [(Dn+2+Mn+2) E2Un+2 (D n+lt+nn+h) thn+h + ...J '- 2R C o o * * D I(Cn+l+ Nn+1) €1Un+l - (Cn*3+Nh+3) e3Uh+3 + ...] 3.1.2h J = Do(l-ClRo) + choRO 3.1.25 K = C1(Ro-DO) + c D 3.1.26 0 1 .111. The remainder of the procedure is a repeated substitution of Eqs. 3.1.23 and 3.1.2h into Eqs. 3.1.16 and 3.1.18 to find expressions for v(£,t) and 1(0,t) in terms of M and N only. The calcuations for evaluat- ing v(£,t) will be given in some detail. which gives V = i o where w * * ‘ The initial step is the expansion of c1 and D1 in Eq. 3.1.16, - 2 - RO(RO Db) N + R00O M U - RO(RO Db) [6' - F. J _ D o D o o D 2 ”h "' 2R 0 2(R -D ) F D R 2 00113-13 .__9_9_. {£211.10 N + D 2 if” D [D 1 D 1 2DlR . ___le [ma/El- 5.4/6.1 ...1 -no [53/21 — 65/61 ...1 i - _. .... .... ... 1 ' %9 [[D3/el ~ DB/el ...] + RO [Ca/€l - cu/el ...] JJ elUl 2R 0 {-{ R K c O O 3 5 7 ° D 5 ‘ D N1 + D"Ml + N1 + 20 -9 0 — z - 3 D [[133/51 - Dr/el ...] + R0 [ta/sl - CMel .. h elUl .- C3 + C5 - C7 00. 3.1.27 "' * 1 28 on = (0n + Nh) EnUn 3. . — *- Dn = (D5 + Mn) enUn 3.1.29 -hg- Equation 3.1.27 is rewritten by collecting terms, expanding the sums J + D and Box + D, and making use of the following two identities: 3.1.31 The resulting expression can be written as 2 v _ RO(Ro-DO) N + R000 M U + 2(RO-Db) Ro(l-ClRo) + DlRo N - £0 ' D o D o o D D M1 D l 2 D D 1 D 1 11 2(R -D) 2DR 2R0 21320 " __9—0— .42- 00. 0° 23$ [13-5 J-R[E-E 1- D D D D D 2 1+... 0 3 5000 2(R -D ) 2(R -D ) 2R 0 0 2J 0 o o O - -' '— '- 3.1.32 It is readily apparent that v in Eq. 3.1.32 is dependent on only 1 o M and N for t < 21, whereas in Eq. 3.1.16 this limited dependence was true only for t'< 1. In general it is found that the expansion of the * * lowest order Cn and Dn will add one time interval 1 to the period over which vz is dependent only on M.and N. Repeated substitution of Eqs. 0 3.1.23 and 3.1.24 to evaluate coefficients is a laborious process, and therefore a recursion fermula for the calculation of coefficients would be most helpful. -113- Tb aid in the development of recursion formulas, the following notation is adopted: 2(RO-DO) $1 = ___—.1) 3.1.33 2ROCO T1 = D 30103“ 2D R 2KR l o o 52 = 51 ——D—- - Tl(1 + T) 3.1.35 2R c 2J o 0 T2 - SI (1 + E") - T1 T 3.1.36 RO (l-ClRO ) DIR: Pn = D Mn + D Nn 3.1.37 R (R -D ) R c O O O O O Qn - ——5——— Nn + D Mn 3.1.38 * * and D in Eq. 3.1.30 leads to the following ex- EXpansion of 02 2 pression for v! : o vzo = noub + [SlPl - TlQl] elUl + [S2P2 - TéQe] €2U2 + 53 :53 - 'D5 ...] -RO [Eu - 66 ...]:l - T3 351; - 1'56 ...] +RO [E3 - 65 ...]:l 3.1.39 where 33 = s2 2D?) T2 (13:19 3.1.ho -uu- 2J 2ROCO 5""T 2 D T = 82 (1 + 3 3.1.“1 Since S3 and T3 are related to 82 and Tb in exactly the same way that 82 and Th are related to S1 and T1’ while Eq. 3.1.39 is related to Eq. 3.1.32 in exactly the same way that Eq. 3.1.32 is related to Eq. 3.1.16, it is possible to write the following general expression for v3 : 0 Vin = v(£,t) = le = QbUb + [slPl - TlQl] elul + [S2P2 - T2Q2] €2U2 + [SnPn - ThQn] enun + : 3.1.h2 where, for n 2 l, 2DlRO 2KRO sn+1 = 5n D - Tn(l + D ) 3.1.u3 2R C 2J o 0 Before the general derivative-explicit equations for case (1) can be written, 1(0,t) must be expressed in terms of.M and N only. The pro- cedure followed is the same as was used for v(£,t), and the final result can be written as iO - 1(0,t) — i o where X1 = Vl = n+1 n+1 O 2=R—Uo*[X1P1'V1Q1]€1U1+ O [X2P2 - V2Q2] «52112 + [XnPn - VnQn] enUn + 2Dl D 2(1-C1RO) D 2DR 21m 1 o o -xn D -vn(1+ D) n_>_1 2RC 2J o o —Xn(l+-I-)-)-Vn D n21 3.1.h5 3.1.1.6 3.1.1.7 3.1.h8 3.1.h9 Substitution of Eqs. 3.1.h2 and 3.1.h5 into Eq. 1.18 ccmpletes the derivation of derivative-explicit equations for case (1). Since the numerous notational changes made in this section tend to obscure the ferm of the results, no direct substitution will be made at this time. However, it may be instructive to consider some general observations which can be made about the derivative-explicit equations for case (1)1: These observations also hold for the Eqs. obtained for the remaining C8863 o -hg- (1) For t < 1, the equations are in normal form. (2) For 1 5 t < 21, a term is added to the equations which reSulted for t < 1. This term involves a linear combination of the functions vC (t-T), i (t-T), e(t-T), and h(t-r). If the equations for tl< 1 haveLEeen solved the linear combination is a specified function and can be grouped with the term.of specified driving functions. Thus the equation for the time interval 1 < t 5.27 has the same mathematical form as Eq. 1.10. (3) For each sucessive interval of duration 1 a similar reasoning can be applied to show that the equations for each interval will have the normal form if the equations for all preceding intervals have been solvedl. (h) Every coefficient for each and every interval is a specified algebraic combination of R0 and submatrices of Eqs. 1.3, l.h, 1.5. In conclusion, the solution of the derivative-explicit equations for RIC networks containing a uniform distortionless transmission line must be carried out in a step-by-step manner due to the reflection associated with the transmission line terminals. In each step of the solution process the equations are in normal form. Reference (1) contains an excellent discussion of the properties of these Eqs., including theorems on the existence of solutions. -h7- 3.2 The general expressions for VT and iT in cases (2), (3) and (h). l 2 The three remaining cases are handled in a manner similar to that used for case (1). The algebraic details are omitted and only the re- sx.ts and notational convention are given. Case (2): neither T element is in the tree, hence the T1 matrices are null. The general expression for i can be written as T2 F— — F-P _ r..— — O 1T == = Q: s U 2 ' l l -i(£,t) _fi; -(le1+le1) -(ann+vnQn) e U n n 3.2.0 where c R R (R +D ) 1 o o o 1 PD _ D Mn - -—-D—-—- Nn 3.2.1 RO(RO+CO) ROCO QD=——-D-———Mn- D Nn 3.2.2 D = (RC-+00) (Ro+Dl) -ch0 3.2.3 20 1 81 = T 302.1l' -h8- T 2(RO+D1) 1 ‘ D x - 2(RO+CO) 1 — D 2Do Vi = “5" 20 R l o Sn+l(xn+l) ' Sn(xn) D - 2K + TD(vn) [1+D—1 m (vm-l) = sn(xn) [Mg-q] + ann) n+1 J = -C0 (RO+D1) + ClDo K = -D1 (RO+CO) + ClDD CO Cl — s' A'1 s - 33 33 D0 D1 N(t) = Sé3A-l [S3Di +535h(t)]+ [Sé3A- M(t) L2 1"]- t_t [ 833A S32G2322 3231 V01 1 S 32 G ' - 2512 O t 813 _52 D ] e(t) + 3.2.5 3.2.6 3.2.7 3.2.8 3.2.9 3.2.10 3.2.11 3.2.12 3.2.13 Case (3): Tb is a branch and T3 is a chord. The general ex- pressions fbr v and i can be written as T T 1 2 1 i=1 VT = v(0,t) = QOU6 + 2E: (SiPi + TiQi) eiUi 3.2.lh -hg- on P . 0 1T2--i(z,t)=Rouo -Zl[xHP +V1Q11611U where R (1 c R ) D R 2 P — 1 ° M + 1 ° N n - D n D n R000 R(Ro D+D) 0 O %= Nn = (l-ROCl) (RO+DD) + ROCODl s _2(RO+DO) 1 ‘ D T _ 2ROC 1 ‘ D X “313.1. 1 ' D v _ 2(l-ClRo) 1 ’ D 2D1Ro 2KRo sn+1 (Xn+l) = 5n (xn ) + TD (vn) [1 + D J ‘ 2R 0 2J o o Tn+1 (Vn+l) ' Sn (Xn) [ET" 1] + Tn (vn) D J = DD (l-ClRo) + choRo K = cl (RO+DD) - CODl 3.2.15 3.2.16 3.2.17 3.2.18 3.2.19 3.2.20 3.2.21 3.2.22 3.2.23 3.2.2h 3.2.25 3.2.26 are null. Case (a): both TD and r . Su3'3u202532A 8. DD = 853A-1 0 = I1 = C1 = SA2FG2 M(t) = vT2 N(t) = iTl The general expression for v r' '-R P O O -R0Qo "U I! I Qn = D = (1+ROD1) (1+aoco) - RO -50- S33 1 3 I '1 I I'1 I I S32A S32G28D2- Geshel + DoiT2 - D + C i - C o Té l S1P1‘T1Q1 lel-vlo‘l T 1 are in the tree, hence the can be written as ' snPn-Tn ' o ann -Vn o N n 2 ClDo 3.2.27 3.2.28 3.2.29 3.2.30 3.2.31 3.2.32 T matrices 2 3-2-35 3.2.36 -51- s - 2Ro cl 1 ' D T _ 2Ro(l+DlRo) 1 ’ D x _ - 2Ro(l+RoCo) 1 ' D 2 v _ 2Ro DD 1 ’ D 20 R 1 0 2K Sn+1 (Xm-l) " ' Sn (Xn) D “Tn (vn) [l + D- r (v )-s (x)[1+?~‘l]+r (v)E-E{5-’-Ecl n+1 n+1 - n n D n n D 2 J = R0 Docl - CORO (1 + RoDl) 2 K = R0 Docl - DlRO (1 + RoCl) o l ._ - 8 '1 I I- - SD2 [ G2832A S32025D2+028D21 D5 D1 N(t) CO Cli = - i + v T M(t) 1 T1 D D o l . 3.3 Some special cases. 3.2.37 3.2.38 3.2.39 3.2.ho 3.2.hl 3.2.h2 3.2.u3 3.2 0141+ 3.2.h5 3.2.u6 General derivative-explicit equations for RIC networks which con- tain a uniform distortionless transmission line of arbitrary length have -52- been derived in the preceding sections. The equations are valid if the transmission line operation is described by Eqs. 1.0 and 1.1 and if cer- tain matrices are non-singular. In general the derivative-explicit equations must be solved in a step-by-step fashion, with new equations, new conditions, and new solu- tions for each time interval 1. The general solution process is lengthy for even simple networks, but there are a number of special cases in which the solution is relatively simple. These special cases are: (l) The network N is disconnected: (a) For all disconnected networks N, one pair of the C i and D1 coefficients is zero in each of the four cases. In case (3), Cl and D5 are zero, while in cases (1), (2), and (1+),CO and D1 are zero. The null coefficients represent a coupling between T6 and T1 in the graph. When N is disconnected, the network graph is disconnected and exactly one of the T elements is in each of the parts of the graph. It can be readily concluded that there is no coupling between T6 and T2 when N is disconnectedl. (b) If N is disconnected and one part of N is resistive, then the terminal quantity (vT or iT ) of Eq. 1.18 which is in the re- 1 2 sistive part of N will not appear in the derivative-explicit equations. l The mathematical basis for this conclusion is contained in footnote 2 on page 8. The truth of this statement will be demonstrated for case (1). Suppose the resistive part of N includes TL in the network graph and the formulation falls into case (1). T is a chord, and Eq. 3.1.3 1 becomes -1 -V = S' A S i 3030 T2 33 33 Té since none of the other terms in Eq. 3.1.3 can exist. A180, 331, B32, and 33% have only zero entries.. By comparing the vanishing coefficients of Eq. 3.1.3 with the coefficients of i in Eq. 1.18, it is fbund that T2 1T will not appear in any of the differential equations. A similar 2 argument can be used if T! is a branch, or if any of the other three cases are investigated. For all disconnected networks N with one resistive part, it is found that either M(t) or N(t) is zero, and either Pn or Qn is zero. For these networks, the resistive part appears in the derivative-ex- plicit equations only through a reflection coefficient. In the special case where the resistive part of the network N is equivalent to R0, this part of the network does not enter the differential equations at all, and the differential-equations take a ferm which describes an RLC net- work in which one of the resistors has the value Rb. Thus the step-by- step solution process is not necessary for this class of networks. (2) The line length l is zero or infinity. (a) l = a; consider the derivative-explicit equations which result for z = m. For finite propagation velocities, we have -5h- 11m T = W 30301 £-€>~ Therefore, U1 5 O, 1 2:1 3.3.2 and the equations of sections 3.1 and 3.2 which specify 1T and vT , in each of the cases, contain a single term. In each of the ibur casts, the single term equation implies that the elemenusTb and T! are equivalent to resistors of resistance R0. The validity of this last statement will be demonstrated for case (1). Eq. 3.1.h5 becomes, for 1 = on, P o i = -—- 3.3.3 T2 Ro Substituting the general forms for P0’ MD, and ND, Eq. 3.3.3 becomes (Ro 1(0,t) - v(O,t)) (l-ClRO) = Dl (RO i(.e,t) + v(z,t)) 3.3.1. For any given network Cl and D1 are specified numbers. Since one of the variables on each side of Eq. 3.3.h can be independently varied, it is concluded that each side must be independently constant. If the in- dependent pair of variables is set equal to zero then the value of the constant is seen to be zero. Since this result is independent of the values of Cl and D1, the terms involving the terminal quantities must be zero. Hence o t O,t = i 3‘3'5 T v I t 1 R0 fifths} = 1'; 3-3-6 -55- Equations 3.3.5 and 3.3.6 are simply defining equations for re- sistors of value R0. (b) I = 0; if derivative-explicit equations have been ob- tained for a network which contains a uniform transmission line and the equations are then examined under the condition of varying line length, the case t = O.must be handled with caution. In general networks, re- ducing the line length to zero may give rise to circuits of voltage drivers, cut-sets of current drivers, or possibly may result in a tOpo- logy such that the tree used as the basis for formulating the equations may no longer conform to the rules specified in Chapter I. If a general network is to be analyzed for the case 1 = O the general procedure should be as follows: (1) Redraw the linear graph for the network with the transmission line removed. (2) Check the resulting graph for conformity with the postulated network restrictions. (3) Formulate and solve the derivative-explicit equations for the graph. While in general the case i = 0 may lead to difficulty, there is at least one class of networks in which I = O:may be considered without reformulation. When N (of Fig. 1.0) is disconnected and one part of N is entirely composed of positive finite resistors, then the case 3 = 0 may be handled by direct substitution of z = 0 into the solutions for vT and iT prior to subsituting these solutions into the equations. 1 2 -56- The validity of this last statement will be shown for case (1), and an example is given in Chapter IV. Assume that the resistive part of N is connected to the x = l port of the transmission line. Since the resistive part of N is simply a two-terminal resistive network, this part can be represented in the graph by a single equivalent resistor Rf. the that 8* is in the cotree for case (1). If the derivative-explicit equations are formulated for the network with special notation used to identify equations relevant to Rf, then the three basic sets of equations can be written in the following form; .1 _ e ‘6 o o o o I s s o s s s " 1c 11 121 13 1h 15 1 I i 0 U o o o I 521 3221 o 323 Set 825 R1 i o 0 U o o , o s o s s D s T 1 321 33 3 35 i =0 3.3.7 o o o 1 o I o o 1 o o o 1.1 I ..... o o o ' i R2 1R. i - T2 11-. h(t) Equation 3.3.7 can be written as . ii- [U : Sc] 1 = 0 3.3.8 -57.. The circuit equations can be written as V 2 I [-SC : U] v II 0 where SC is given by Eqs. 3.3.7 and 3.3.8. The element equations are the same as in equation 1.5 except for the cotree resistors. The representation used is i G 0 R2 2 = 1 i O — * *- In the matrix equations, G2, R2 R2 * resistors except R , and the matrix 812 of Chapter I becomes [S in the equations in this section. 12 3.3.10 v , and 1 refer to all cotree If the various coefficients of Chapter III are evaluated for the case under consideration, the results are D1 = C0 = 0 l c =-— *- l R -l — - ' A S DB S33 33 3.3.11 3.3.12 3-3-13 3.3.1h 3-3-15 -58- Qn .... 0 3.3.16 2KR R - R l + D O = T___O. 3°3°l7 R + R o R +D 2J o o l + b— - Ro-D 303.18 It can be shown that Do is a negative definite quadratic fern, and .2139 R0 - D0 Substitution of t = 0 into the equations fer v(£,t) and 1(0,t) therefore < 1 303019 yields: Q i v(z,t) l = P0 N Z (KoKl) ) 3.3.20 i = 0 i=0 2x ° 1 o i 1(0,t) l = P0 fi— + fi< Z (KOKI) ) 3.3.21 3 = O O O 0 i=0 where 2R* T = W 303022 *- RO-R Kb = §;:§x' 303.23 0 = 3.3.214- K1 Ro-Do -59.. P = M 303025 Since IKDKll < l, the infinite series converge. Using the result that i 1 Z (KOKl) .-. 1-K K 3.3.26 i=0 0 l the expressions for the terminal quantities can be written as * v(£,t) ' = —§—— MO 3.2.27 .2 = O R -DO MO 1(0,t) = * 3.3.28 fl = O R -Db From Eq. 3.1.13, Mo = v(O,t) --DO i(O,t) 3.3.29 Solving Eq. 3.3.29 for v(O,t) and evaluating for t = 0 gives v(O,t) = MD + DO 1(0,t) 1 3.3.30 3 = 0 42 = 0 From.Eqs. 3.3.27 and 3.3.28, v(OIt) = V(IIt) I 303031 2 = 0 l = 0 From.Eq. 3.1.10, since N6 = O, V T1 1 = - —— 3°3'32 T1 R* -60- Since iT = - i(f,t), Eq. 3.3.32 yields 1 i(£,t) = 1(0,t) 3.3-33 z = 0 z = 0 It can be concluded that substitution of I = 0 into the expressions for v(£,t) and i(O,t) has had the effect of replacing the element TD by * , . the resistor R , and has not violated Kirchhoff's laws. IV. EXIMPLES OF THE ANALYSIS h.O Introduction In the preceding chapters, derivative-explicit equations have been derived for RLC networks which contain a uniform distortionless transmission line. Several special cases in which the solution of the equations is relatively simple have been mentioned in Chapter III. In section h.1, an example problem which illustrates the general formulation and several special cases is considered. In section n.2, the formulation of a necessary and sufficient set of eguations is discussed. In section h.3, the alternative formulation discussed in Chapter I is illustrated. h.l An example of the analytical procedure. The network of Figure h.l.0 will be analyzed by using the de- rivative-explicit formulation presented in the preceding chapters. The choice of the network used in this example was based on the ease with which special cases can be illustrated as the general analysis is per- formed. The following features of the network should be noted: (1) N is disconnected, (2) One part of N is resistive. -61- () 0 C) Figure h.l.0 The example network. Since N is disconnected, the two-port graphical representation of the transmission line is known to be valid. Figure h.l.l gives the oriented linear graph which is used as the basis for the formulation. The voltage driver and initial conditions are specified by the following equations: e(t) = 1, t > 0 #10 e(t) = O, t < O vC(O) = 0 1+.l.l iL(O) = 0 u.1.2 'fhc initial step in the analysis is the selection of a tree by application of the rules in section 1.1. The tree must include e(t), C, and either T, or R. In this example Ti is selected as a branch to make £ \. VR T2 _. L1. Figure h.1.l Oriented linear graph for the example network. the formulation fit into case (1) since this case has been more thoroughly detailed than the other cases in preceding sections. The f-cut-set, f-circuit, and element equations are, respectively: e(t) c T, R To L rie ‘ 1 '1 0 0 I 0 1 1 “ iC l . 2 0 1 0 0 -1 -1 1T! = O h-lo3 . l h .0 0 1 , l 0 0 '{“' ’ R 2 3 h i : TO 3.11: 4 I ' e(t)“ 2 ”'0 0 -1 . 1 0 0 v I c 3 -l 1 0 : 0 l 0 VTD = 0 h.1.u I ..... h -1 1 0 I 0 0 1.. VR 1 2 h ‘ VT I O . -‘ ’— O "' F _ [1C C 0 VC 0 ‘ : hole 1R O G 0 vR 5 v 0 0 L 1 {LLJ a _ _Ll where the integers below and to the left of Eqs. h.l.3 and h.1.h in- dicate the j and i of S and B13 as these submatrices appear in the 13 preceding chapters. The dot over the entries in the right hand column matrix of Eq. h.l.5 denotes the time derivative. Substitution of the appropriate submatrices into Eq. 1.18 yields the modified derivative explicit equations as . 7 -l " ' F - " — F _ — ' vC C 0 O 1 VC 0 O l VT! 0 = + 8(t)+ “0106 LIL-J 0 L :1 0‘31” _ld LO 0- 3T0— Performing the indicated matrix operations in Eq. h.l.6 gives t =$[i +1 ] 1+.l.7 C C L T o L L C Note that VT does not appear in the modified derivative-explicit £ equations. This result is in accord with the discussion in section 3.3. The next step in the analysis is the calculation of i from.Eq. Th 3.1.h5. The various coefficients are evaluated from their respective defining equations by substitution from Eqs. h.1.3, h.1.h and h.1.5. The results are CO = D0 = D1 = O 1‘0109 1 Cl = " fi' Ll-.l.10 N = 0 h.1.11 n Mn = (e(t-nw) - vC(t-n1)) U(t-n1) h.l.l2 -65- D = R0 (1 + ? holol3 Pn = Mn )-|-.l.ll+ Q.“ = 0 [#01015 J = 0 #116 R0 K = - fi— halal? x1 = o 1+.1.18 2 Vl = fi— h01019 o Ro-R xn+1 = vn (DE-1;) h.1.20 = X holo2l n+1 n In accordance with the conclusions reached in section 3.3, note Since the coefficient of Vn in Eq. h.l.20 has the general form of a current reflection coefficient, the following notation will be used in the remainder of this example: K = 0 1+.1.22 Based on the value of the coefficients calculated for this example, iT can be written as 0 -66- 1 = i: [e(t)-v6] + g_ X x: [e(t-Zi 1) - vc (t-ei 1)] U(t-2i r) T R o O 131 “01023 The derivative-explicit equations for the example network can now be written in expanded form as: . 1 [ 1 2K6 VC = 5- 1L + fiz-(e(t)-vc(t)) + -§;-(e(t-21) - vc(t-21)) U(t-21) 2x: + -§- (e(t-ht) - vc(t-h1)) U(t-hr) + ...] h.l.2h 0 1L . 31: (e(t) - vc(t)) h.1.8 Before considering the general solution of Eqs. h.l.2h and h.l.8 these equations will be examined under some of the special circumstances discussed in section 3.3. (1) If I = a, the derivative-explicit equations become vc = 3% [1L + i: (e(t) - vc(t))] u.1.25 iL = % (e(t) - vc(t)) h.1.26 Equations h.1.25 and h.1.26 describe the network of Fig. h.l.2. In essence, Tb has been replaced by a resistor Rb. > i e(t) L R Figure 4.1.2 Reduced example network and graph for special cases. -67- (2) If R = R0 Eq. h.l.22 indicates Kb = 0. Thus the derivative- explicit equations are Eqs. h.l.25 and h.l.26 for this case. (3) If 2 = 0, then 1 = O, and Eqn. h.l.2h can be written as [H [1L - <- II can; a: O (e(t)-vc(t)) + g. (e(t)-vC(t)) jg: xi] u.1.27 ° i=0 where %- (e(t) - vc(t)) has been added and subtracted inside the bracket. 0 When [Kb] < l, the sum.in Eqn. h.l.27 converges: G i l . 0 i=0 Thus, when 0 0, Eq. h.l.36 can be written in terms of the differential operator D as D l 2 1 (D + R00 + 122—)Vc ‘ ‘15 #437 The roots of the characteristic equation are -.;L_.+'\/Q_£_92 - E_j ROC - RoC LC D = 2 h.1.38 Three cases can occur with respect to the roots. The particular case which does arise is determined by the value of (fiifi’z - %Ew The cases 0 are: 1 2 h (1) Roots real and distinct for (— > —. ROC LC 1 2 h (2) Roots are complex conjugates for (-——- <-—— RbC LC. -70- (3) Roots are real and repeated for (-l'—-2 = E_, RoC LC The values assigned below to the network elements for this example give rise to real repeated roots. R = 1 ohm o C = l farad L = h henrys R = % ohm 1 = 1 second From Eq. h.l.22 and the specified values of R and R0, K5 is evaluated as l K ... 5 “.39 v = Kl e + K: t e + l h.l.h0 where K1 and K? are constants which can be evaluated by using the initial conditions given in Eqs. h.l.l and h.l.2. The solution obtained for v c can be written as -2 2 t vc=e (5-1).»1 ogt<2 h.1.l+1 Substituting Eq. h.1.1+1 into Eq. h.1.3l+ gives -.‘9. t 2 -71- At t = 2 another term in Eq. 4.1.2h becomes non-zero. The dif- ferential equations for 2 S t < 1+ are vC = 1L + l - vC + l - vC(t-2) h.l.h3 1 2 5 t < 1+ I: [l - VC] Ll'olol'l‘h In the time interval for which Eq. h.l.h3 is valid, the time delayed function is obtained from Eq. h.l.hl as gt-22 2 vC(t-2) = e - (5&3 - 1) + 1 2 g t < h h.1.l+5 The conditions imposed to evaluate constants of integration which arise in the solution of Eqs. h.l.h3 and h.l.hh are that v and i C L must be continuous at t = 2. The complete solutions for Eqs. h.l.h3 and h.l.hh are t l -- -—(t-2) 3 2 2 t 2 1: 3t 7 13 vc=e (§-l)+l+e (ET-T+§-§—) 25t<1+ holoh6 t l —— -—(t-2) 3 2 iL=E e2+e2 (fig-}+%§——§) 2_<_t + $- (l-vc A LB iLB -l l 1 -RA '0 0.. i O 0 Th + -l O 1 T2 0 O After obtaining and substituting the solution’for Chapter IIIfmanipulation of Eq. h.2.lO yields 1 in case (2) from Th e(t) h.2.lO where f(vC (t-n1)) represents the time delayed terms in i C hrth If!“ H 2m sF1*‘ oh: 0 JP“ JI>SU ><‘)|l*-l :51)?! OOIP UOIH T -81- 0 £1" 0 uf’l“ O f(vcc(t-n1')) 0 0 O ”Fla? 35"" #51:? 00'“ afl", _ Th 0 :51", O J'l" e(t) + lI.2.ll Since Eq. h.2.ll is in the form of Eq. 1+.2.l, Eqs. lI.2.lt and 152.6 can be used to obtain 7 >0le 0 O O J'IJ' ...: I >°|w° O O ashlar We (0)“ vc (0) i .A B. 1. (o) a?» A 0 LV ”o ' F— L — '- '- 1 .A v o o - -—(1+——) v Ca Ca ts on L ' 1 1 .A v . o - --—(i.——) v Cc Rocc cc LB 00 . c 1 B 1 1 1 1 -—{1+-—) -—- -R (-—-+ -—) 1 __PA__ [PA CA. PA ‘A FA LB_, L_;h J ' L 0 TL 0 "d" 0 0 393 s L eta c .03 +R_A -..].T... ..R_A -l_. 0 L_¥AQA ts FA 9A 9A _ — q ch(O) 11‘ 0 ch< ) 1LB(0) e(t) f(vC (t-n1)) i 0) L C d 14.2.13 Equation h.l.13 specifies a necessary and sufficient set of de- rivative-explicit equations for the network of Fig. h.2.o. h.3 An example of the alternate formulation. In Chapter I the possibility of selecting a tree by rules other than those postulated was mentioned. The conclusion reached was that 'while a somewhat different system of derivative-explicit equations could be obtained, there was no possibility of reducing the number of equations in the system by the alternate formmlation. The second possible tree is based on the following rules: (1) All e(t) elements are in the tree. -83- (2) .All possible C elements are placed in the tree, unless Tb and/or T: is in parallel with any of these capacitors. In this case, either terminal elements or the parallel capacitors may be put into the tree. (3) All possible L elements are placed in the cotree, unless Tb and/or T: is in series with any of these inductors. In this case, either the terminal elements or the series inductors may be placed in the cotree. (h) All h(t) elements are in the cotree. By considering the network of Fig. h.3.l, this alternative forumla- tion is briefly illustrated. The oriented linear graph used in this example is the graph of Fig. h.l.l with the C and L elements interchanged. Since C is now in parallel with To, a tree composed of e(t), R, and TB will be used in the formulation. __-- ‘v Figure h.3.0 Emample network for illustration of alternative fbrmulation. The corresponding f-cut-set, f-circuit, and element equations are, respectively: different set of variables as analogous to Eq. 1.12 is e(t) lo If each of the vectors c)! -an- h.3.o h.3.l h.3.2 of Eq. h.3.2 is expressed in terms of a was done in Chapter I, the equation O1 o] The modified derivative-explicit equations obtained from Eq. h.3.3 are l l VT 0 E VT 0 'c d ° ° [ I [1 I. I. 633- 1 = -£’ 0 1 + E. e(t) + 0 1TIo .3. L L L L The results obtained in section h.2 imply that the rank of the * coefficient matrix Bb determines the number of necessary and sufficient simultaneous equations for a given network. The normal form for Eq. h.3.h can be expressed as —T - fi— % vT O l e(t) at ‘ - + “3'5 - l 1 3L - i o iL f o f(vTo(t-n'r)) * and since Bb has full rank, the necessary number of equations is two. Therefore, no reduction in the number of equations results from the al- ternate formulation. V. CONCLUSIONS 5.0 Conclusion In the first three chapters of this thesis, systems of derivative- cxglicit equations have been derived fer RLC networks containing a uni- form, distortionless, transmission line. The solution of the equations has been shown to be sufficient to complete the solution of the network. The general form of the derivative-explicit equations is a series of time-delayed terms which become sequentially non-zero. The solution of the equations must be obtained by a step-by-step process since the equations for various time intervals involve the solutions for previous time intervals. The formulation of a minimum order set of equations from the general sufficient set could conceivably result in a significant saving of labor. Since networks do not ordinarily contain series capacitors nor parallel inductors, the real significance of the reduced set formula- tion would be where component values are responsible for the "excess" equations. 5.1 Additionalqproblems. At least two t0pics worthy of further investigation have arisen in this thesis. (1) In Chapter I, the two-port representation was only shown to -86- -87- be valid when N is disconnected. It would be very helpful if all net- works in which the two-port representation is valid could be character- ized. One possible starting point for such an investigation could be the consistency of results obtained when the transmission line is re- placed by a generalized equivalent circuit. The equivalent circuit it- self would have to be consistent with Eqs. 1.0 and 1.1. (2) The appearance of time-delayed terms in the derivative- explicit equations has a rather general effect on the solutions of the equations. Examination of Eq. h.2.0 leads to the conclusion that the characteristic equations will be the same for every time interval. In every time interval after the first, the characteristic roots will appear in the term f(t) since f(t) contains time-delayed terms which corres- pond to solutions for previous intervals. Calculation of the particular integrals will therefore result in polynomials which increase in order as the solution is carried out for additional time intervals. In general, the rate of increase of the polynomial order is related to the multi- plicity of the characteristic roots. The solutions obtained for v and C iL in section h.l illustrate these points. A further investigation into the polynomials might lead to relationships which would allow computer calculation of the polynomials. This result would yield an appreciable reduction in labor if solutions were to be obtained for a large number of time intervals. APPENDIX A SOME THEOREMS AND DEFINITIONS Theorem A(l): (2, p. 98)1 If the columns of the f-circuit and f-cut- set matrices of an oriented graph G are arranged in the same order of branches and chords for a defining tree so that [U i s 1 I B = [B : U] and Sf I C Theorem A(2): If the square matrix A of order n has rank r Eln, and if the matrix A is written as A11 A12 A = A21 A22 -1 where All is of order r and non-singular, then A22 - A21 A11 A12 — O. Parenthesis give reference and page number where equivalent, or more general, theorems or defs. are stated. -88- -89- Proof: The matrix A can be factored as indicated in the identity A.l. . _l ‘ A11 A12 U 0 A11 0 U A11A'12 E -1 "l A. 1 A21 A22 A21"‘11 U 0 A22 "A21A11A12 O U The rank of the matrix product must be r. Since the first and last matrices in the product are non-singular, the rank of the matrix in the middle of the product must be r. This conclusion is based on the fact that when an arbitrary matrix B is pre-(or post-) multiplied by a conformable non-singular matrix, the product has the rank of B (11, p. 109). By hypothesis, has rank r, and hence the first r rows of the middle A11 matrix are linearly independent. Since 0 submatrices appear above and -l to the left of the submatrix A22 - A21 A11 A12, any non-zero entry in A22 - A21 Aii A12 will result in a row which is linearly independent of the first r rows of the matrix. This violates the hypothesis, and there- -1 fore A22 ' A21 A11 A12 = 0' Definition A(l): (2, p. 89) The vertex matrix A8' of an oriented graph is defined by Aa = [aij] is of order v x e for a graph with v vertices and e elements, where aij = I if element 3 is incident at vertex i and is oriented away from vertex i aij = -1 if element J is incident at vertex i and is oriented toward vertex i, and aij = 0 if element 3 is not incident at vertex i. Definition A(2): (10, p. 256) A real symmetric matrix A is called a positive definite matrix if and only if the corresponding quadratic form X'AX is positive definite. 10. ll. -91- REFERENCES Wirth, J. L., Time Domain Models of Physical Systems and Existence of Solutions, Ph.D. Thesis, Michigan State University, 1962. Seshu, S. and Reed, M. B., Linear Graphs and Electrical Networks, Addison-Wesley Company, 1961 Brown, D. P., "Derivative-Explicit Differential Equations for RLC Graphs," JOurnal of the Franklin Institute, Volume 275, pp. 503-51h (1963)- Brown, R. G., Sharpe, R. A., and Hughes, W. L., Lines, Waves, and Antennas, Ronald Press Company, 1961 Potter, J. L., and Fich, S. J., Theory of Networks and Lines, Pren- tice-Hall, Inc., 1963. Ku, Y. H., Transient Circuit Analysis, D. Van Nestrand Company, Inc., 1961. Koenig, H. E., and Blackwell, w. A., Electromechanical System Theory, McGraw-Hill Book Co., 1961. Goldman, 8., Transformation Calculus and Electrical Transients, Pren- tice-Hall Inc., l9h9. Campbell, G. A., and Foster, R. M., Fourier Integrals for Practical Applications, D. Van Nestrand Company, Inc., l9h8. Bonn, F. E., Elementary_Matrix Algebra, The MacMillan Company, 1958. I Cramer, H., Mathematical Methods of Statistics, Princeton University Press, l9h6. ROOM USE 0:41: 3% 7: :ii' I in] 4009 IIIIIIIIIIIII’I‘I’ L IIIIIIIIIIIIIIIIIIIIIIII