AN .ABSTRACT OF THE THESIS OF Edward Janes Swenson for the M. S. in Electrical Engineering (Name) (Degree) (Major) Date thesis is presented June 30, 1966 ----------------------~~~------------------ Title HIGH RESOLUTION ROOT LOCUS PLOTS --------------~------~~~----~~--~~--~~-------------- Abstract approved Redacted for Privacy --·~o~----~·~·----~(M--a~jo_r__p-ro~f~e-ss-o~r~)------------------ A technique is presented for obtaining high resolution root locus plots and Bode diagrams. This is done by translating the origin of coordinates to the area of interest by an appropriate change of variable. The method is applicable to hand calculations or use of a potential analog computer. Examples include the analyses of systems that contain very highly damped, closely spaced pole zero pairs. These are actual design problems encountered in the design of high-speed tracking antenna systems and booster control systems. HIGH RESOLUTION ROOT LOCUS PLOTS by EDWARD JAMES SWENSON A THESIS submitted to OREGON STATE UNIVERSITY in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE June 1967 HIGH RESOLUTION ROOT LOCUS PLOTS by EDWARD JAMES SWENSON ©Copyright 1966 by Electro Scientific Industries, Inc. Litho U.S.A. APPROVED: Redacted for Privacy Associate Professor of Electrical Engineering In Charge of Major Redacted for Privacy Hedd/of Department of Elbctrital and Electronics Engineering ( Redacted for Privacy Dean of Graduate Schoo I Date thesis is presented June 30, 1966 ------------~------------- Typed by Loraine Riley ACKNOWLEDGMENT The author wishes to express appreciation to Professor Solon A. Stone for his constructive suggestions offered during the preparation of this thesis. TABLE OF CONTENTS I. The Log s Plane 1 1.1 Introduction . 1 1.2 The Log s Plane 2 1.3 Need for Scale Expansion 5 II. Origin Translation in the Log s Plane 7 2. 1 Basic Equat·ions 7 2.2 Origin Translation 8 2.3 Axis Location 9 Ill. Real Axis Translation 13 3. 1 Introduction 13 3. 2 Roots Near Pole or Zero 13 3.3 Angle Breakaways 17 IV. Root Locus Analysis for Closely Spaced Complex Pairs 21 4. 1 Translation Along jw Axis . 21 4.2 Translation in the Plane .... 24 4.3 Bode Diagrams of Resonant Pairs . 25 v. Compensation Synthesis in Log x Plane 36 5. 1 Introduction . 36 5.2 Damping Ratio Template 38 5. 3 Design Example . 41 VI. Origin Translation in the z Plane 53 6. 1 Introduction . 53 6. 2 The z Plane and Log z Plane 53 6.3 Log z Plane Origin Translation 56 6.4 Cone luding Comments . 57 Bibliography . 59 Appendix A: Changing Factor Forms 61 Appendix B: Generalized Frequency Response Plots 65 FIGURES 1. 1 The s and log s planes in rectangular and polar coordinates 3 1.2 Danping ratio lines in the s and log s plane 4 2. 1a Translation along the negative real axis •• 10 2. 1 b Translation along the positive real axis •• 10 2.2 Translation along the imaginary axis 11 2.3 Translation in a complex direction 12 3.1 System block diagram · • . • • 13 3.2 Root locus plot of Equation 3.2 • 14 3.3 Root locus plot of Equation 3.4 15 3. 4 Root locus plot of Equation 3. 13 • 19 3.5 Real axis plot or "Siggy" plot of Equation 3.13 20 4.1 Root locus plot of Equation 4.1 • • . • • • •••••••• 22 4.2 Log x plane root locus plot of Equation 4.1 (st = +j30) • • ••• 24 4.3 Log x plane root locus plot of Equation 4.1 (st = -1 + jlOO) •• 26 4.4 Phase plot of Equation 4.1 • • • • • • • • • • • • • • 27 4.5 GainplotofEquation4.1 • . • • . • • . • • • 28 4 .6a log x plane phase plot of Equation 4.1 for w positive (90° line) •• 29 4 .6b log x plane phase plot of Equation 4. 1 for w negative ( 270° line) • 30 4. 7 Phase plot of Equation 4. 1 with the log x plane information added • 31 4.8a log x plane gain plot of Equation 4.1 for w positive . • • . 33 4.8b Log x plane gain plot of Equation 4.1 for w negative . 34 4.9 Gain plot of Equation 4.1 with the log x plane information added • 35 5.1 Damping ratio lines in the log s plane . • . • • • 36 5.2a The s plane dcrnping ratio and undamped natural frequency lines plotted on the log x plane (at positive) . • . 37 5.2b The s plane damping ratio and undanped natural frequency lines plotted on the log x plane (at negative) • . • • • . 38 5.3 The s plane damping ratio and undamped natural frequency lines plotted on the log x plane (Wt positive) . • . • . • . • 39 5.4 The s plane dcrnping ratio and undamped natural frequency lines plotted on the log x plane (at negative and Wt positive) • 40 5.5 Template for constructing damping ratio lines • • • • • •• 40 5.6 Calibrated damping ratio template ( DRT) • 41 5.7 System block diagram . • • 41 5.8 Root locus plot of Equation 5.2 • . • 42 5.9 Root locus plot of Equation 5.2 with lead network compensation added . • . • • • . • . • . • . • 43 5. 10 log x pI ane root locus plot of zero pole pairs x3 - x 13 and x5-x15with lead network z1 -P1 added (st=+j42.1)... 45 5.11 log x plane root locus plot of zero pole pair x7- x17 with lead network Z1- P1 added (st = +j85.1) . • . 46 5.12 log x plane root locus plot of zero pole pair x9- x19 with lead network zl- pl added (st=+201.4) .............. 47 5.13 Symmetrical twin-tee or parallel-tee network ...•..••.• 48 5.14 log x plane root locus plot of zero pole pair X9- x19 with lead network zl - pl and twin-tee network p2 - p3 and z2- z3 added . 50 5.15 Root locus plot of Equation 5.2 with lead network Z1 - P1 and twin-tee network P2 - P3 and Z2- Z3 added ••......•. 51 5.16 Root locus plot of Equation 5.2 W,ith lead network and twin-tee compensation added . : . • . • . 52 6. 1 The s plane, z plane and log z plane . • . • . 54 6.2a The log z plane with s plane damping and undamped natural frequency lines (positive real axis in center) . • . 55 6.2b The log z plane with s plane damping and undamped natural frequency lines (negative real axis in center) . • . • 56 6.3 Figure 6.2a when translated to the z1 plane (z = z1 + 1) . 57 6.4 Figure 6.2b when translated to the z 1 plane (z = zl + 1) 58 TABLES 4. 1 Translating the origin to +j30 ............ 23 4.2 Translating the origin to -1 + jlOO ... 25 4.3 Transforming Figures 4 .6a and 4 .6b to the s plane . 31 4.4 Transforming Figures 4.8a and 4.8b to the s plane ..... 32 5. 1 Translating the origin to +j42.1, +j85.1 and +j201.4 . 44 HIGH RESOLUTION ROOT LOCUS PLOTS CHAPTER I THE LOG s PLANE 1 . 1 Introduction As automatic control systems have become increasingly complex through the years, new and more powerful techniques for designing and analyzing them have been required. Although studies in the time domain using digita I and ana­ log computers do provide answers to systems whose performance has been we II specified, transform methods are extremely important in the design of systems with a number of variable parameters. The transform methods have been partic­ ularly useful for design since they allow a wide variety of performance criteria and visualization aids to be applied. The most widely used of the visualization aids are the Nyquist Diagram, Nichols Chart, Bode Plot and Root Locus Plot. Of the four graphical methods, the most popular by far has been the Bode plot. Its popularity stems from its simplicity of construction and its ability to represent all of the singularities of the transfer function on one plot. Diffi­ culties with the Bode diagram are encountered, however, if several quadratic terms appear in the transfer function. In recent years the root locus method has become popular because it overcomes the difficu Ities encountered with the Bode diagram in that it com­ pletely specifies the variation of the closed loop pole locations as the open loop 2 gain, or some other parameter, is varied. Normally, root locus plots are plotted on a linear coordinate system as originally developed by Evans (3, 4). The s plane was adopted because of its familiarity and ease of calculation. Unfortu­ nately, however, there are several disadvantages with the s plane root locus plot. These are: (1) Accuracy is difficult to obtain for a system with widely separated roots. (2) Gain versus frequency and system band width (which may be specified} are not apparent. (3} The effect of sma II gain changes in the system performance cannot be pre­ dicted readily. 1 . 2 The Log s Plane The difficulties encountered with the root locus plot can be overcome if the log s plane is used since: (1} All the poles and zeros of the transfer function can be represented with equa I resolution and accuracy over every decade. (2) The system band width is completely specified and can be sketched directly from the log s plane root locus plot because the root locations line up directly with the break frequencies of the standard Bode diagram. (3} The effect of small gain changes can be observed, particularly at lower fre­ quencies. Figure 1 . 1 shows the s plane and log s plane for both rectangular and 3 s PLANE LOG s PLANE 90• 270• a:>­ crz c:; cr :IE + 0z cr cr..J +j "'a: + Figure 1.1 The s and log s planes 1n rectangular and polar coordinates.