ILL I N 0 I S UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN PRODUCTION NOTE University of Illinois at Urbana-Champaign Library Large-scale Digitization Project, 2007. UNIVERSITY OF ILLINOIS BULLETIN IsSSD TWICE A WEEK Vol. XXXV November 30, 1937 No. 27 [Entered' as second-class matter December 11, 1912, at the- post office at Urlana, Illinois, under the Act of August 24, 1912. Acceptance for mailing at the special rate of postage provided for in section 1103. Act of October 3, 1917, authorized July 31, 1918.] SOLUTION OF ELECTRICAL NETWORKS BY SUCCESSIVE APPROXIMATIONS BYLARENCEL. SMITH LAURENCE L. SMITH BULLETIN No. 299 ENGINEERING EXPERIMENT STATION Pssmn pr Tws UNvZRaST OFILLmNOIs. UBAiA ?AoPE: Fowrr-nvtCTk T HE Engineering Experiment Station was established by act of the Board of Trustees of the University of Illinois on De- cember 8, 1903. It is the purpose of the Station to conduct investigations and make studies of importance to the engineering, manufacturing, railway, mining, and other industrial interests of the State. The management of the Engineering Experiment Station is vested in an Executive Staff composed of the Director and his Assistant, the Heads of the several Departments in the College of Engineering, and the Professor of Industrial Chemistry. This Staff is responsible for the establishment of general policies governing the work of the Station, including the approval of material for publication. All members of the teaching staff of the College are encouraged to engage in scientific research, either directly or in co6peration with the Research Corps, composed of full-time research assistants, research graduate assistants, and special investigators. To render the results of its scientific investigations available to the public, the Engineering Experiment Station publishes and dis- tributes a series of bulletins. Occasionally it publishes circulars of timely interest, presenting information of importance, compiled from various sources which may not readily be accessible to the clientele of the Station, and reprints of articles appearing in the technical press written by members of the staff. The volume and number at the top of the front cover page are merely arbitrary numbers and refer to the general publications of the University. .ither above the title or below the seal is given the num- ber of the Engineering Experiment Station bulletin, circular, or reprint which should be used in referring to these publicatiohs. IFor copies of publications or for other information address THE ENGINEERING EXPERIMENT STATION, IfNIVASITYr oF IlNOIS, URBANA, ILLINOIS 1 1 1 * I, * : ' - ' -^ '*' , " ; .. ; " :' ^ ' '* *. UNIVERSITY OF ILLINOIS ENGINEERING EXPERIMENT STATION BULLETIN No. 299 NOVEMBER, 1937 SOLUTION OF ELECTRICAL NETWORKS BY SUCCESSIVE APPROXIMATIONS BY LAURENCE L. SMITH ASSOCIATE IN ELECTRICAL ENGINEERING ENGINEERING EXPERIMENT STATION PUBLISHED BY THE UNIVERSITY OF ILLINOIS, URBANA 3000-12-37-13321 PRESS , CONTENTS PAGE I. INTRODUCTION . 5 1. Introductory . 5 2. Acknowledgments . 5 II. DISCUSSION OF METHODS OF SOLUTION . 5 3. Methods Available . 5 4. Solution by Calculating Tables . 6 5. Solution by Simultaneous Equations . 6 6. Solution by Trial and Error . 7 7. Solution by Simplification of Network . 8 8. Solution by Method of Successive Approximations 8 III. METHOD OF BALANCING VOLTAGE DROPS . 10 9. Description . 10 10. Illustrative Problem-Single d-c. Loop . 10 11. Illustrative Problem-Two-Loop d-c. Circuit 12 IV. METHOD OF BALANCING CURRENTS . 14 12. Description . 14 13. General Proof . 14 14. Illustrative Problem-Single d-c. Loop 20 15. Illustrative Problem-Two-Loop d-c. Circuit 20 16. Illustrative Problem-Three-Loop d-c. Circuit 21 17. Illustrative Problem-Modification of Method Used in Section 16 . 24 18. Illustrative Problem-Partial Excess Method 25 V. SOLUTION OF TYPICAL NETWORK PROBLEM . 27 19. Statement of Problem . 27 20. Solution by Simultaneous Equations . 28 21. Solution by Balancing Voltage Drops . 29 22. Solution by Balancing Currents . 29 VI. CONCLUSIONS . 33 23. Method of Balancing Voltage Drops . 33 24. Method of Balancing Currents . 33 APPENDIX A . 35 Suggested Forms . 35 APPENDIX B . 38 Bibliography . 38 LIST OF FIGURES NO. PAGE 1. Envelope Curve for Approximations-Load Current . 31 2. Envelope Curve for Approximations-Input Current . 31 3. Envelope Curve for Approximations-Network Current . 32 SOLUTION OF ELECTRICAL NETWORKS BY SUCCESSIVE APPROXIMATIONS I. INTRODUCTION 1. Introductory.-The calculation of voltage and current condi- tions in electrical networks becomes increasingly important as central station systems grow and their proper operation becomes more im- portant to insure continuity of service to each load carried. Voltage regulation, current division between paths, transmission efficiencies, and selective operation of relays must be known for the present sys- tem or for the system resulting after proposed additions to the present system, either under normal operation or under short circuit condi- tions. Unfortunately, as the importance of calculating such con- ditions increases, so also does the difficulty of making such calculations increase. It is the purpose of this bulletin to present an additional method of solving these networks. Other methods are available, but it is felt that this method has considerable merit, particularly when applied to the solution of a-c. secondary networks or the usual transmission line systems. Although the method, as presented in this paper, is applied to a-c. and d-c. power networks, it should be borne in mind that the method is applicable to any network composed of invariable pa- rameters which are constant with respect to time. The discussion of the advantages and limitations of this method will be confined to a comparison with other existing methods used in d-c. or a-c. power network problems. 2. Acknowledgments.-The writer wishes to express his apprecia- tion for the assistance and suggestions given by PROF. HARDY CROSS and MR. L. B. ARCHER in the preparation of this manuscript. II. DIscussioN OF METHODS OF SOLUTION 3. Methods Available.-Various methods of attack in obtaining the solution of electrical networks have been devised and are in use. These methods may be grouped into five general classifications: (a) Solution by calculating tables (b) Solution by simultaneous equations (c) Solution by trial and error (d) Solution by simplification of the network (e) Solution by method of successive approximations 5 ILLINOIS ENGINEERING EXPERIMENT STATION 4. Solution by Calculating Tables.-Calculating tables* were orig- inally devised by H. H. Dewey and W. W. Lewis. These tables were groups of variable calibrated resistances with plug-in leads by which any electrical network could be duplicated by a proper set-up of resistances on the table. A d-c. voltage was applied at the generating source, and the resultant currents read in each branch of the network. The value of the resistance inserted in each branch was determined by the value of the reactance in that particular branch of the actual network being studied. The resistance of the actual network was neglected. These tables were mainly used in finding short-circuit conditions. Their use in studying normal operating conditions was slight, since phase displacements of generated voltages and impedance angles of the network could not be duplicated by such a d-c. table. A-c. calculating tablest are an extension of the principles used in the d-c. calculating tables, in that the actual system is duplicated in the laboratory by means of the calculating table. It consists of many variable resistors, inductances, and capacitances, so that each unit of the actual network may be accurately duplicated, and these units combined into a network such as the one to be studied. An a-c. voltage is applied, and the currents in each branch are measured. Phase shifters may be used to simulate the phase displacements in voltages of the alternators on the actual system. Short-circuit conditions or normal operating conditions may be studied on this type of set-up, since the actual system is very exactly duplicated. Its accuracy is mainly limited by the evaluation of the constants in the actual system. The chief disadvantage of the calculating table method is its cost. Few companies, though much interested in network problems, could justify the purchase of such a table. As a result, few tables are now available, and many problems worthy of solution are not solved by this method. 5. Solution by Simultaneous Equations.-The solution of a net- work by simultaneous equations is so generally known that little reference to the method need be made. The simultaneous equations are set up to satisfy the two fundamental laws of electrical circuits, known as Kirchhoff's Laws. The first law states that the total cur- *G. E. Rev., p. 901, 1916. G. E. Rev., Vol. 22, p. 140, 1919. G. E. Rev., p. 669, 1920. Elec. World, p. 985, 1922. Elec. World, p. 707, 1923. tTrans. A. I. E. E., p. 10, 1923. G. E. Rev., p. 611, 1923. Trans. A. I. E. E., p. 831, 1923. Trans. A. I. E. E., p. 72, 1925. SOLUTION OF ELECTRICAL NETWORKS rent flowing into each junction is zero. The second law states that the total drop in voltage around each closed loop is zero. To deter- mine the voltage drop in a particular branch, Ohm's Law as applied to alternating current is used. This law states that the voltage drop across an impedance is equal to the product of the current flow through it and the value of the impedance. The set of simultaneous equations, so obtained, may be solved to obtain the current flowing in each branch. If the number of equations is large, this process is an extremely tedious and cumbersome one. The process may be shortened by the use of determinants, but the calculations are diffi- cult to check, and the intermediate steps in the solution do not have a physical significance that is easy to grasp. 6.