Calculation of the Two High Voltage Transmission Line Conductors Minimum Distance
|
|
- Bathsheba Bradley
- 6 years ago
- Views:
Transcription
1 World Journal of Engineering and Technology, 15, 3, Published Online Ocober 15 in SciRes. hp:// hp://dx.doi.org/1.436/wje.15.33c14 Calculaion of he Two High Volage Transmission Line Conducors Minimum Disance Wenjie Huang 1, Jinglin Zhu 1, Zhihang Du 1, Zheng Zhang 1 Shanghai Elecric Power Design Insiue co., LTD, Shanghai, China Economic and Technology Research Insiue of Shanghai Elecric Power Company, Shanghai, China huangwj@sepd.com.cn, @qq.com Received 9 Augus 15; acceped 15 Ocober 15; published Ocober 15 Absrac In he design of high volage ransmission lines ineviably needed o change he arrangemen of wires, and he disance beween he wires direcly affeced by changing he arrangemen of he wires. The disance beween he wires is difficul o judge by experiences. Therefore, i is urgenly o develop a way o accuraely calculaing he minimum disance beween he wires when changing he arrangemen beween he wo wires, and deermine he minimum disance can mee he requiremens of sandards and regulaions or no. Through he hinking, based on he balance equaion and he space curve wire calculus mehod, a more accurae mehod of calculaing he wo wires in a variey of condiions he minimum spacing was derivaed, and he sensiiviy of he minimum disance was analysized based on he mehod. Keywords Overhead Transmission Lines, Disance of Two Lines, Equaion of Wire Balance, Wire Sag 1. Inroducion When he high volage ransmission lines are designed, i is ineviably needed o inersec across, change he arrangemen modes of wires, and ec., for example, 5 kv line spans he kv line, ge-in ganry span of overhead lines, and ec. When he above siuaions happen, i is ineviably needed o check he minimum disance beween wo wires so as o deermine wheher i saisfies he requiremen of regulaion [1]. However, he difficulies of validaion lie in (1) he inersecion siuaion, which needed o be checked, is ofen very complex, and he wo wires o be checked face he siuaions such as differen ypes, differen safey coefficiens, and so on. () Boh he esablishmen and soluion of he space curve equaion are difficul, and relying on a simple arificial modeling calculaion canno obain he exac soluion. (3) The spacings of wires are differen due o he differen sags under differen working condiions, so i is difficul o judge he working condiion for he minimum spacing, and ec. There are much calculaing sudy achievemen concerning he phase spacing of wires in our counry. In he Reference [], he hree-dimensional CAD is used o esablish he hree-dimensional model of phase spacing of lead-in span wire, and a kind of mehod o calculae he phase spacing of wires by using he inerac- How o cie his paper: Huang, W.J., Zhu, J.L., Du, Z.H. and Zhang, Z. (15) Calculaion of he Two High Volage Transmission Line Conducors Minimum Disance. World Journal of Engineering and Technology, 3, hp://dx.doi.org/1.436/wje.15.33c14
2 ive hree-dimensional CAD according o he heory of oblique parabola and hree-dimensional lef conversion calculaion is proposed. Reference [3] sudied he compuing mehod of minimum phase spacing of wires under considering he windage and wihou considering he windage, and analyzed he calculaion examples of relevan projecs. According o he above difficulies and he curren saus, space curve equaion [4]-[6] of wires was precisely esablished and he minimum disance beween wo wires was calculaed by using he inegral-differenial mehod of space curve and Malab ool in consideraion of he sag of wires under differen working condiions based on he equaion of wire balance. Simulaneously, he accuracy of he calculaion mehod was analyzed, and he sensiiviy of minimum disance beween wo wires wih he change of ower heigh difference, he change of safey coefficien, he change of span, and so on was sysemaically calculaed by aking he energy-saving wire (aluminum alloy core aluminum sranded conducor) as an example.. Calculaion Mehod of Spaial Minimum Disance beween Two Wires.1. Esablishmen of Space Curve Equaion of Wires The emphasis of calculaing he minimum disance beween wo wires in space is o esablish he calculaion model of space curve of wires. Firs of all, he hree-dimensional recangular coordinae sysem is esablished by aking he ower cener as he original poin, he exension direcion of beam as X axis, upward exension direcion of ower as Z axis, and he exension line direcion of wire as Y axis, as shown in Figure 1. Figure 1. Case diagram of modeling calculaion. The space curve equaion was esablished by aking wire 1 as an example. I is assumed ha hanging poins coordinaes in he T 1 and T ower of he wire are (x 1, y 1, z 1 ) and (x, y, z ), respecively, and hen he space curve parameer equaion of wire 1 beween he wo suspension poins is as follows [5]. x = x1 + ( x x1) y = y1 + ( y y1) z = z1 + ( z z1) where is he spaial locaion parameers. l The wire sag was also considered, and he sag formula of inclined parabola of wire was f = γ according o reference [4], hen he sag of any poin in he space wire 8σ cos β is 9
3 γ ( l x') x' f = σ cos β where f is he wire sag wih he uni m. γ is he comprehensive relaive load of wire wih he uni N/mm m. l is he horizonal disance of wire beween wo hanging poins wih a uni m. x ' is he disance from any poin of wire o suspension poin (x 1, y 1, z 1 ) wih a uni m. β is he angle of elevaion difference wih a uni. Then he calculaion mehod of he above various parameers can be deermined according o he model. l = ( x x1) + ( y y1) x' = ( x x1) + ( y y1) = ( x + ( x x ) x ) + ( y + ( y y ) y ) = l L= (( x x1) + ( y y1) + (z z1) cos β = l / L The calculaed simplified formula of he above parameers are inroduced ino he sag formula, hen ( ) γ Ll f = σ.. Correcion of Spaial Wire Equaion in Consideraion of Windage There appears a cerain windage for he wire under he acion of breeze, and is angle of windage is ϑ= arcan γ4 / γ 1, hen sin ϑ= γ4 / γ and cos ϑ= γ1 / γ can be obained. Afer he windage of wire, he change of sag in Z axis is z = f cosϑ, and he change of sag on he horizonal plane is xy = f sinϑ. I is assumed ha he sraigh line linking wih he suspension poins of wire and he X axis in XY plane consiues an angle α, and hen α = arcan y x y y y y y x x x x , sinα = =, cosα = = x 1 ( x x1) + ( y y1) l ( x x1) + ( y y1) l Due o he windage direcion of wire along X axis unfixed, when he direcions of windage are differen, he variable quaniy in X and Y axis of he line beween wo hanging poins caused by wire sag are shown in Table 1 in consideraion of synchronous windage. Hence, when he wire has he windage along posiive direcion of X axis, he space equaion of wire can be expressed as: γ Ly ( y1)( ) x = x1 + ( x x1) + σ γ Lx ( x1)( ) y y1 ( y y1) = + σ γ Ll( ) z = z1 + ( z z1) σ While he wire has he windage along negaive direcion of X axis, he space equaion of wire can be obained Table 1. Variaion in X and Y axis caused by sag under differen direcions of windage. Direcion of windage Variable quaniy Windage along posiive direcion of X axis Windage along negaive direcion of X axis Change along X axis Change along Y axis /m X /m Y γ Ly y σ ( )( ) 1 ( )( ) 1 γ Lx x σ γ Ly y σ ( )( ) 1 γ Ly y σ ( )( ) 1 91
4 by exchanging he plus-minus sign of above sag correcion formula..3. Elecrical Disance beween Two Wires in Space The parameer equaion of space wire has been obained in consideraion of he sag and windage according o he above analysis. I is assumed ha he wo wires in space are line1 and line, respecively, he hanging poins coordinaes of line1 are A (x 1, y 1, z 1 ) and B (x, y, z ), respecively, and he hanging poins coordinaes of line are C (x 3, y 3, z 3 ) and D (x 4, y 4, z 4 ), respecively. Then he disance beween any poins in wo wires can be calculaed according o he disance formula beween wo poins in space. γ L( y4 y3)( ) d = [ x3 + ( x4 x3) + σ γ L1( y y1)( 1 1 ) x1 ( x x1) 1 ] σ γ L( x4 x3)( ) + [ y3 + ( y4 y3) σ γ L1( x x1)( 1 1 ) y1 ( y y1) 1 + ] σ γ Ll ( ) + [ z3 + ( z4 z3) σ γ Ll 11( 1 1 ) z1 ( z z1) 1 + ] σ Visibly, d is he binary quaric equaion, and he minimum of d needs o be resolved. According o he differenial mehod, le d / 1 =, d / =, and hen he binary cubic equaion se concerning he posiion funcion 1 and is obained. Subsequenly, he Malab plaform can be used o solve d, hus he minimum of d is obained. The binary cubic equaion has muliple ses of soluions, so he range of funcion values 1 [ y1, y], [ y3, y4] needs o be defined. Moreover, 1 and appear in pairs, d has a unique soluion afer he definiion of funcion values, and i can be quickly solved in Malab environmen. 3. Sensiiviy Analysis of Minimum of Spaial Two Wires When he arrangemen mode of high volage overhead lines changes, he elecrical spacing of wire mus be verified. Especially, i is commonly seen ha he ge-in ganry span wire is convered from he horizonal layou of ganry o he verical arrangemen form ouside of a saion. I is analyzed ha he sensibiliy of minimum disance beween wo wires wih he change of disance beween erminal ower and ganry, he change of heigh difference beween erminal ower and ganry, change of safey coefficien of relaxaion span, and so on by aking he ge-in ganry span of kv lines engineering and adoping he JL1/LHA1-465/1 (aluminum alloy core aluminium conducor) Model Analysis The model of sensiiviy analysis is shown in Figure []. Where he ype of he wire is JL1/LHA1-465/1, and he parameers are shown in Table. 3.. Change Law of Minimum Disance beween Two Wires under Differen Spans 1) Change law of disance wih he span under differen working condiions According o he sipulaions of GB , he minimum phase spacings in span under he acion of inerphase operaing overvolage a 5 kv and he volage grade below 5 kv are shown in Table 3. The regulaion only sipulaes he minimum disance of phase spacing of wire under operaing condiions, and he minimum phase spacing of wire is. m under kv. Bu he minimum disance beween wires does no necessarily happen under operaing condiions. Hence, i is necessary o compare he disance beween wires under high emperaure o ha under operaing condiions, as shown in Figure 3. 9
5 Figure. Model of sensiiviy analysis. Table. Parameers of energy-saving wire. Type of he wire Aluminum alloy core aluminium conducor Seel (Aluminium-coaed seel) 9.85 Secional area (mm ) Aluminum (Aluminium alloy) Toal cross secion Aluminium/seel (Aluminium-coaed seel) cross-secion raio. Diameer (mm) Uni weigh (kg/km) Comprehensive elasic coefficien (MPa) 55 Comprehensive expansion coefficien (1/ C) Calculaed ensile force (kn) 137. Tension-weigh raio (T/W) 7.49 Table 3. The minimum inerphase inervals in span under he acion of inerphase operaing overvolage. Nominal volage/kv Inerphase inerval (in span)/m Figure 3. Comparison of disances beween wires under differen working condiions. 93
6 As shown in Figure 3, he minimum disance of wire under he operaing condiions is obviously greaer han he minimum disance under he high emperaure condiions. Moreover, when he minimum span is m, he minimum phase spacings under boh operaing condiions and high emperaure condiions can mee he requiremens of regulaion. Bu he minimum disance of wires under he operaing condiions canno represen oher condiions o sudy he minimum disance of wires. Therefore, i is likely o use he minimum disance of wire under high emperaure condiions o sudy he minimum disance of wires. ) Change law of disance wih span under he high emperaure In he calculaion model, he wo hanging poins coordinaes of wire 1 are (7,, 7) and (3.5, y, 14), respecively, as well as he hanging poins coordinaes of wire are (7.8,, 33.5) and (, y 4, 14), respecively. In addiion, he difference of hanging poins disance beween wire 1 and wire in he erminal ower is 6.5 m. Figure 4. Change law of minimum spacing beween wo wires under differen spans and differen safey coefficiens. I is clearly observed ha (1) Under he idenical safey coefficien, he disance beween wo wires in space increases wih he increase of he span. However, he ampliude of increase coninuously decreases as well as he disance ends o be smooh and seady. Taking k = 15 as an example, when he span is in - 3 m, he rae of change of he increase of he shores disance is abou 5.5%. When he span is in 11-1 m, he change rae is only.7%. Namely, he greaer he span is, he weaker he conrol of minimum disance beween wo wires in space by he wire ension is. () Wih he increase of safey coefficien, he change law of he shores disance beween wo wires in space wih he span becomes more and more obvious, as well as he rae of change is greaer. As k = 3.4, he change rae of span in - 1 m is 3.8%, while k = 15, he rae of change is 15.3% wih he same span. So when designing he change of he arrangemen mode for wires, i is quie necessary o check wheher he space disance beween wo wires mee he requiremens. (3) Under he idenical span and differen safey coefficiens, i is observed ha he wires relax and he sag of wires increases wih he increase of securiy coefficiens. These make he minimum disance beween wo wires coninuously reduce. (4) I is required in he regulaion ha he inerphase disance of kv wires is no less han m. Bu only considering he operaion saus in he engineering is no enough, i is sill considered o mee he requiremen of m for he phase spacing under he high emperaure, moreover, he phase spacing should be increased by % under he allowable condiions. In his case, he phase spacing should no be less han.4 m. When he wires are relaxing, he safey coefficien and he span should be simulaneously conrolled, for example, as k = 15, he span should no be less han 3 m, while k =, he span should no be less han 5 m Change Law of Minimum Disance wih he Span under Differen Heigh Differences When designing he ge-in/ou ganry span, i is ofen encounered ha he difference beween he nominal heigh of erminal ower and he heigh of ganry is greaer. Hence, he change of he minimum disance of wo wires in space wih differen heigh differences needs o be sudied. So when k = 15, he change of he minimum disance in space beween wo wires of ge-in ganry span needs o be iniially accouned by designing Δh = 13 m, m, 3 m, 4 m and ec. 94
7 Figure 5. Change law of minimum disance beween wo wires in space wih he span under differen heigh differences. As can be seen ha: (1) Wih he increase of he heigh difference beween he erminal ower and ganry, he minimum spacing beween wo wires are coninuously reducing, and he greaer he heigh difference is, he greaer he shorened ampliude is. Moreover, when Δh is in 13 m - m, m - 3 m, and 3 m - 4 m, he corresponding rae of change is.8%, 1.5%, and 1.94%, respecively, wih he span of 6 m. () Wihin he idenical span, he minimum disance beween wo wires are coninuously shorening wih he increase of he heigh difference. However, as he span becomes greaer, he shores disance beween wo wires ends o be consisen. Namely, he greaer he span is, he weaker he conrol of he shores disance beween he wo wires by heigh difference is. 4. Conclusions Based on he equaion of wire balance, space curve equaion of wires was precisely esablished and he minimum disance beween wo wires in space was calculaed by using he inegral-differenial mehod of space curve and Malab ool in consideraion of he sag of wires under differen working condiions. Simulaneously, he sensiiviy of minimum disance beween wo wires wih he change of ower heigh difference, he change of safey coefficien, he change of span, and so on was sysemaically calculaed by aking he energy-saving wire (aluminum alloy core aluminium conducor) as an example. Moreover, he conclusions are obained as follows. (1) Under he same safey coefficien, he disance beween wo wires in space increases wih he increase of span, bu he increase ampliude gradually decreases as well as he disance ends o be sable. Moreover, he greaer he span is, he weaker he conrol of he minimum disance beween he wo wires in space by he change of wire ension caused by he change of securiy coefficien of wire is. () Under he same span, he sag of he wire increases wih he increase of he safey coefficien, which makes he minimum disance beween he wo wires coninuously decrease. (3) Wih he increase of ower heigh difference beween boh sides, he minimum spacing beween wo wires gradually decreases, and he greaer he heigh difference is, he greaer he shorened ampliude is. (4) Wihin he same span, he minimum disance beween wo wires gradually shorens wih he increase of he heigh difference. Bu he greaer he span is, he weaker he conrol of he shores disance beween wo wires by he heigh difference is. References [1] GB Code for Design of 11kV - 75kV Overhead Transmission Line. [] Yang, X.M., Feng, N. and Wu, S.P. (1) Precise Calibraion for Phase Disance of Approach Span Based on Three- Dimensional Space Technology. Elecric Power Consrucion, 33, [3] Bai, X.L., Ge, Q.L. and Yu, W.W. (1) Analysis of Leas Phase Disance of Overhead Transmission Line. Elecric Power Science and Engineering, 6,
8 [4] Zhang, D.S. (3) Design Handbook for Elecric Power Transmission Line. China Power Press, Beijing. [5] Gong, Y.Q., Liang, N.C. and Gong, Y.G. (9) A Formula Mehod for Conducor Elecric Clearance of Overhead Transmission Line. Elecric Power Consrucion, 3, 4-7. [6] Rohlfs, A.F. and Schneider, H.M. (1983) Swiching Impulse Srengh of Compaced Transmission Line Fla and Dela Configuraions. IEEE Transacions on Power Apparaus and Sysems, 1,
1. VELOCITY AND ACCELERATION
1. VELOCITY AND ACCELERATION 1.1 Kinemaics Equaions s = u + 1 a and s = v 1 a s = 1 (u + v) v = u + as 1. Displacemen-Time Graph Gradien = speed 1.3 Velociy-Time Graph Gradien = acceleraion Area under
More informationIB Physics Kinematics Worksheet
IB Physics Kinemaics Workshee Wrie full soluions and noes for muliple choice answers. Do no use a calculaor for muliple choice answers. 1. Which of he following is a correc definiion of average acceleraion?
More information15. Vector Valued Functions
1. Vecor Valued Funcions Up o his poin, we have presened vecors wih consan componens, for example, 1, and,,4. However, we can allow he componens of a vecor o be funcions of a common variable. For example,
More informationLAPLACE TRANSFORM AND TRANSFER FUNCTION
CHBE320 LECTURE V LAPLACE TRANSFORM AND TRANSFER FUNCTION Professor Dae Ryook Yang Spring 2018 Dep. of Chemical and Biological Engineering 5-1 Road Map of he Lecure V Laplace Transform and Transfer funcions
More informationLecture 2-1 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still.
Lecure - Kinemaics in One Dimension Displacemen, Velociy and Acceleraion Everyhing in he world is moving. Nohing says sill. Moion occurs a all scales of he universe, saring from he moion of elecrons in
More informationIn this chapter the model of free motion under gravity is extended to objects projected at an angle. When you have completed it, you should
Cambridge Universiy Press 978--36-60033-7 Cambridge Inernaional AS and A Level Mahemaics: Mechanics Coursebook Excerp More Informaion Chaper The moion of projeciles In his chaper he model of free moion
More informationSolutions from Chapter 9.1 and 9.2
Soluions from Chaper 9 and 92 Secion 9 Problem # This basically boils down o an exercise in he chain rule from calculus We are looking for soluions of he form: u( x) = f( k x c) where k x R 3 and k is
More informationHall effect. Formulae :- 1) Hall coefficient RH = cm / Coulumb. 2) Magnetic induction BY 2
Page of 6 all effec Aim :- ) To deermine he all coefficien (R ) ) To measure he unknown magneic field (B ) and o compare i wih ha measured by he Gaussmeer (B ). Apparaus :- ) Gauss meer wih probe ) Elecromagne
More information!!"#"$%&#'()!"#&'(*%)+,&',-)./0)1-*23)
"#"$%&#'()"#&'(*%)+,&',-)./)1-*) #$%&'()*+,&',-.%,/)*+,-&1*#$)()5*6$+$%*,7&*-'-&1*(,-&*6&,7.$%$+*&%'(*8$&',-,%'-&1*(,-&*6&,79*(&,%: ;..,*&1$&$.$%&'()*1$$.,'&',-9*(&,%)?%*,('&5
More information2.1: What is physics? Ch02: Motion along a straight line. 2.2: Motion. 2.3: Position, Displacement, Distance
Ch: Moion along a sraigh line Moion Posiion and Displacemen Average Velociy and Average Speed Insananeous Velociy and Speed Acceleraion Consan Acceleraion: A Special Case Anoher Look a Consan Acceleraion
More informationModule 2 F c i k c s la l w a s o s f dif di fusi s o i n
Module Fick s laws of diffusion Fick s laws of diffusion and hin film soluion Adolf Fick (1855) proposed: d J α d d d J (mole/m s) flu (m /s) diffusion coefficien and (mole/m 3 ) concenraion of ions, aoms
More informationConstant Acceleration
Objecive Consan Acceleraion To deermine he acceleraion of objecs moving along a sraigh line wih consan acceleraion. Inroducion The posiion y of a paricle moving along a sraigh line wih a consan acceleraion
More informationKINEMATICS IN ONE DIMENSION
KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings move how far (disance and displacemen), how fas (speed and velociy), and how fas ha how fas changes (acceleraion). We say ha an objec
More informationSimulation-Solving Dynamic Models ABE 5646 Week 2, Spring 2010
Simulaion-Solving Dynamic Models ABE 5646 Week 2, Spring 2010 Week Descripion Reading Maerial 2 Compuer Simulaion of Dynamic Models Finie Difference, coninuous saes, discree ime Simple Mehods Euler Trapezoid
More informationLecture 4 Kinetics of a particle Part 3: Impulse and Momentum
MEE Engineering Mechanics II Lecure 4 Lecure 4 Kineics of a paricle Par 3: Impulse and Momenum Linear impulse and momenum Saring from he equaion of moion for a paricle of mass m which is subjeced o an
More informationMath 333 Problem Set #2 Solution 14 February 2003
Mah 333 Problem Se #2 Soluion 14 February 2003 A1. Solve he iniial value problem dy dx = x2 + e 3x ; 2y 4 y(0) = 1. Soluion: This is separable; we wrie 2y 4 dy = x 2 + e x dx and inegrae o ge The iniial
More informationLinear Response Theory: The connection between QFT and experiments
Phys540.nb 39 3 Linear Response Theory: The connecion beween QFT and experimens 3.1. Basic conceps and ideas Q: How do we measure he conduciviy of a meal? A: we firs inroduce a weak elecric field E, and
More informationTraveling Waves. Chapter Introduction
Chaper 4 Traveling Waves 4.1 Inroducion To dae, we have considered oscillaions, i.e., periodic, ofen harmonic, variaions of a physical characerisic of a sysem. The sysem a one ime is indisinguishable from
More informationChapter 2. First Order Scalar Equations
Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.
More informationWEEK-3 Recitation PHYS 131. of the projectile s velocity remains constant throughout the motion, since the acceleration a x
WEEK-3 Reciaion PHYS 131 Ch. 3: FOC 1, 3, 4, 6, 14. Problems 9, 37, 41 & 71 and Ch. 4: FOC 1, 3, 5, 8. Problems 3, 5 & 16. Feb 8, 018 Ch. 3: FOC 1, 3, 4, 6, 14. 1. (a) The horizonal componen of he projecile
More informationVehicle Arrival Models : Headway
Chaper 12 Vehicle Arrival Models : Headway 12.1 Inroducion Modelling arrival of vehicle a secion of road is an imporan sep in raffic flow modelling. I has imporan applicaion in raffic flow simulaion where
More information23.5. Half-Range Series. Introduction. Prerequisites. Learning Outcomes
Half-Range Series 2.5 Inroducion In his Secion we address he following problem: Can we find a Fourier series expansion of a funcion defined over a finie inerval? Of course we recognise ha such a funcion
More informationReasonable compensation coefficient of maximum gradient in long railway tunnels
Journal of Modern Transporaion Volume 9 Number March 0 Page -8 Journal homepage: jm.swju.edu.cn DOI: 0.007/BF0335735 Reasonable compensaion coefficien of maximum gradien in long railway unnels Sirong YI
More informationMath 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities:
Mah 4 Eam Review Problems Problem. Calculae he 3rd Taylor polynomial for arcsin a =. Soluion. Le f() = arcsin. For his problem, we use he formula f() + f () + f ()! + f () 3! for he 3rd Taylor polynomial
More information1. The graph below shows the variation with time t of the acceleration a of an object from t = 0 to t = T. a
Kinemaics Paper 1 1. The graph below shows he ariaion wih ime of he acceleraion a of an objec from = o = T. a T The shaded area under he graph represens change in A. displacemen. B. elociy. C. momenum.
More information2.7. Some common engineering functions. Introduction. Prerequisites. Learning Outcomes
Some common engineering funcions 2.7 Inroducion This secion provides a caalogue of some common funcions ofen used in Science and Engineering. These include polynomials, raional funcions, he modulus funcion
More informationHamilton- J acobi Equation: Weak S olution We continue the study of the Hamilton-Jacobi equation:
M ah 5 7 Fall 9 L ecure O c. 4, 9 ) Hamilon- J acobi Equaion: Weak S oluion We coninue he sudy of he Hamilon-Jacobi equaion: We have shown ha u + H D u) = R n, ) ; u = g R n { = }. ). In general we canno
More informationSolution: b All the terms must have the dimension of acceleration. We see that, indeed, each term has the units of acceleration
PHYS 54 Tes Pracice Soluions Spring 8 Q: [4] Knowing ha in he ne epression a is acceleraion, v is speed, is posiion and is ime, from a dimensional v poin of view, he equaion a is a) incorrec b) correc
More informationPhysics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle
Chaper 2 Newonian Mechanics Single Paricle In his Chaper we will review wha Newon s laws of mechanics ell us abou he moion of a single paricle. Newon s laws are only valid in suiable reference frames,
More informationTHE 2-BODY PROBLEM. FIGURE 1. A pair of ellipses sharing a common focus. (c,b) c+a ROBERT J. VANDERBEI
THE 2-BODY PROBLEM ROBERT J. VANDERBEI ABSTRACT. In his shor noe, we show ha a pair of ellipses wih a common focus is a soluion o he 2-body problem. INTRODUCTION. Solving he 2-body problem from scrach
More informationSub Module 2.6. Measurement of transient temperature
Mechanical Measuremens Prof. S.P.Venkaeshan Sub Module 2.6 Measuremen of ransien emperaure Many processes of engineering relevance involve variaions wih respec o ime. The sysem properies like emperaure,
More information18 Biological models with discrete time
8 Biological models wih discree ime The mos imporan applicaions, however, may be pedagogical. The elegan body of mahemaical heory peraining o linear sysems (Fourier analysis, orhogonal funcions, and so
More informationEchocardiography Project and Finite Fourier Series
Echocardiography Projec and Finie Fourier Series 1 U M An echocardiagram is a plo of how a porion of he hear moves as he funcion of ime over he one or more hearbea cycles If he hearbea repeas iself every
More informationThe motions of the celt on a horizontal plane with viscous friction
The h Join Inernaional Conference on Mulibody Sysem Dynamics June 8, 18, Lisboa, Porugal The moions of he cel on a horizonal plane wih viscous fricion Maria A. Munisyna 1 1 Moscow Insiue of Physics and
More informationThe Paradox of Twins Described in a Three-dimensional Space-time Frame
The Paradox of Twins Described in a Three-dimensional Space-ime Frame Tower Chen E_mail: chen@uguam.uog.edu Division of Mahemaical Sciences Universiy of Guam, USA Zeon Chen E_mail: zeon_chen@yahoo.com
More informationECE 2100 Circuit Analysis
ECE 1 Circui Analysis Lesson 35 Chaper 8: Second Order Circuis Daniel M. Liynski, Ph.D. ECE 1 Circui Analysis Lesson 3-34 Chaper 7: Firs Order Circuis (Naural response RC & RL circuis, Singulariy funcions,
More information3, so θ = arccos
Mahemaics 210 Professor Alan H Sein Monday, Ocober 1, 2007 SOLUTIONS This problem se is worh 50 poins 1 Find he angle beween he vecors (2, 7, 3) and (5, 2, 4) Soluion: Le θ be he angle (2, 7, 3) (5, 2,
More informationReading from Young & Freedman: For this topic, read sections 25.4 & 25.5, the introduction to chapter 26 and sections 26.1 to 26.2 & 26.4.
PHY1 Elecriciy Topic 7 (Lecures 1 & 11) Elecric Circuis n his opic, we will cover: 1) Elecromoive Force (EMF) ) Series and parallel resisor combinaions 3) Kirchhoff s rules for circuis 4) Time dependence
More informationGround Rules. PC1221 Fundamentals of Physics I. Kinematics. Position. Lectures 3 and 4 Motion in One Dimension. A/Prof Tay Seng Chuan
Ground Rules PC11 Fundamenals of Physics I Lecures 3 and 4 Moion in One Dimension A/Prof Tay Seng Chuan 1 Swich off your handphone and pager Swich off your lapop compuer and keep i No alking while lecure
More informationDifferential Equations
Mah 21 (Fall 29) Differenial Equaions Soluion #3 1. Find he paricular soluion of he following differenial equaion by variaion of parameer (a) y + y = csc (b) 2 y + y y = ln, > Soluion: (a) The corresponding
More informationV L. DT s D T s t. Figure 1: Buck-boost converter: inductor current i(t) in the continuous conduction mode.
ECE 445 Analysis and Design of Power Elecronic Circuis Problem Se 7 Soluions Problem PS7.1 Erickson, Problem 5.1 Soluion (a) Firs, recall he operaion of he buck-boos converer in he coninuous conducion
More informationInventory Analysis and Management. Multi-Period Stochastic Models: Optimality of (s, S) Policy for K-Convex Objective Functions
Muli-Period Sochasic Models: Opimali of (s, S) Polic for -Convex Objecive Funcions Consider a seing similar o he N-sage newsvendor problem excep ha now here is a fixed re-ordering cos (> 0) for each (re-)order.
More information23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More informationSome Basic Information about M-S-D Systems
Some Basic Informaion abou M-S-D Sysems 1 Inroducion We wan o give some summary of he facs concerning unforced (homogeneous) and forced (non-homogeneous) models for linear oscillaors governed by second-order,
More informationRC, RL and RLC circuits
Name Dae Time o Complee h m Parner Course/ Secion / Grade RC, RL and RLC circuis Inroducion In his experimen we will invesigae he behavior of circuis conaining combinaions of resisors, capaciors, and inducors.
More informationStability and Bifurcation in a Neural Network Model with Two Delays
Inernaional Mahemaical Forum, Vol. 6, 11, no. 35, 175-1731 Sabiliy and Bifurcaion in a Neural Nework Model wih Two Delays GuangPing Hu and XiaoLing Li School of Mahemaics and Physics, Nanjing Universiy
More informationAt the end of this lesson, the students should be able to understand
Insrucional Objecives A he end of his lesson, he sudens should be able o undersand Sress concenraion and he facors responsible. Deerminaion of sress concenraion facor; experimenal and heoreical mehods.
More informationClass Meeting # 10: Introduction to the Wave Equation
MATH 8.5 COURSE NOTES - CLASS MEETING # 0 8.5 Inroducion o PDEs, Fall 0 Professor: Jared Speck Class Meeing # 0: Inroducion o he Wave Equaion. Wha is he wave equaion? The sandard wave equaion for a funcion
More information( ) ( ) if t = t. It must satisfy the identity. So, bulkiness of the unit impulse (hyper)function is equal to 1. The defining characteristic is
UNIT IMPULSE RESPONSE, UNIT STEP RESPONSE, STABILITY. Uni impulse funcion (Dirac dela funcion, dela funcion) rigorously defined is no sricly a funcion, bu disribuion (or measure), precise reamen requires
More informationEXERCISES FOR SECTION 1.5
1.5 Exisence and Uniqueness of Soluions 43 20. 1 v c 21. 1 v c 1 2 4 6 8 10 1 2 2 4 6 8 10 Graph of approximae soluion obained using Euler s mehod wih = 0.1. Graph of approximae soluion obained using Euler
More informationGuest Lectures for Dr. MacFarlane s EE3350 Part Deux
Gues Lecures for Dr. MacFarlane s EE3350 Par Deux Michael Plane Mon., 08-30-2010 Wrie name in corner. Poin ou his is a review, so I will go faser. Remind hem o go lisen o online lecure abou geing an A
More informationNavneet Saini, Mayank Goyal, Vishal Bansal (2013); Term Project AML310; Indian Institute of Technology Delhi
Creep in Viscoelasic Subsances Numerical mehods o calculae he coefficiens of he Prony equaion using creep es daa and Herediary Inegrals Mehod Navnee Saini, Mayank Goyal, Vishal Bansal (23); Term Projec
More informationBasic Circuit Elements Professor J R Lucas November 2001
Basic Circui Elemens - J ucas An elecrical circui is an inerconnecion of circui elemens. These circui elemens can be caegorised ino wo ypes, namely acive and passive elemens. Some Definiions/explanaions
More informationQ.1 Define work and its unit?
CHP # 6 ORK AND ENERGY Q.1 Define work and is uni? A. ORK I can be define as when we applied a force on a body and he body covers a disance in he direcion of force, hen we say ha work is done. I is a scalar
More informationLet us start with a two dimensional case. We consider a vector ( x,
Roaion marices We consider now roaion marices in wo and hree dimensions. We sar wih wo dimensions since wo dimensions are easier han hree o undersand, and one dimension is a lile oo simple. However, our
More information6.01: Introduction to EECS I Lecture 8 March 29, 2011
6.01: Inroducion o EES I Lecure 8 March 29, 2011 6.01: Inroducion o EES I Op-Amps Las Time: The ircui Absracion ircuis represen sysems as connecions of elemens hrough which currens (hrough variables) flow
More informationcopper ring magnetic field
IB PHYSICS: Magneic Fields, lecromagneic Inducion, Alernaing Curren 1. This quesion is abou elecromagneic inducion. In 1831 Michael Faraday demonsraed hree ways of inducing an elecric curren in a ring
More informationInventory Control of Perishable Items in a Two-Echelon Supply Chain
Journal of Indusrial Engineering, Universiy of ehran, Special Issue,, PP. 69-77 69 Invenory Conrol of Perishable Iems in a wo-echelon Supply Chain Fariborz Jolai *, Elmira Gheisariha and Farnaz Nojavan
More informationMath 116 Practice for Exam 2
Mah 6 Pracice for Exam Generaed Ocober 3, 7 Name: SOLUTIONS Insrucor: Secion Number:. This exam has 5 quesions. Noe ha he problems are no of equal difficuly, so you may wan o skip over and reurn o a problem
More informationNumerical Dispersion
eview of Linear Numerical Sabiliy Numerical Dispersion n he previous lecure, we considered he linear numerical sabiliy of boh advecion and diffusion erms when approimaed wih several spaial and emporal
More informationChapter 1 Electric Circuit Variables
Chaper 1 Elecric Circui Variables Exercises Exercise 1.2-1 Find he charge ha has enered an elemen by ime when i = 8 2 4 A, 0. Assume q() = 0 for < 0. 8 3 2 Answer: q () = 2 C 3 () 2 i = 8 4 A 2 8 3 2 8
More information4. Electric field lines with respect to equipotential surfaces are
Pre-es Quasi-saic elecromagneism. The field produced by primary charge Q and by an uncharged conducing plane disanced from Q by disance d is equal o he field produced wihou conducing plane by wo following
More information1 Differential Equation Investigations using Customizable
Differenial Equaion Invesigaions using Cusomizable Mahles Rober Decker The Universiy of Harford Absrac. The auhor has developed some plaform independen, freely available, ineracive programs (mahles) for
More informationPhysics Equation List :Form 4 Introduction to Physics
Physics Equaion Lis :Form 4 Inroducion o Physics Relaive Deviaion Relaive Deviaion Mean Deviaion 00% Mean Value Prefixes Unis for Area and Volume Prefixes Value Sandard form Symbol Tera 000 000 000 000
More informationMotion along a Straight Line
chaper 2 Moion along a Sraigh Line verage speed and average velociy (Secion 2.2) 1. Velociy versus speed Cone in he ebook: fer Eample 2. Insananeous velociy and insananeous acceleraion (Secions 2.3, 2.4)
More informationChapters 6 & 7: Trigonometric Functions of Angles and Real Numbers. Divide both Sides by 180
Algebra Chapers & : Trigonomeric Funcions of Angles and Real Numbers Chapers & : Trigonomeric Funcions of Angles and Real Numbers - Angle Measures Radians: - a uni (rad o measure he size of an angle. rad
More informationUnsteady Flow Problems
School of Mechanical Aerospace and Civil Engineering Unseady Flow Problems T. J. Craf George Begg Building, C41 TPFE MSc CFD-1 Reading: J. Ferziger, M. Peric, Compuaional Mehods for Fluid Dynamics H.K.
More informationCLASS XI SET A PHYSICS. 1. If and Let. The correct order of % error in. (a) (b) x = y > z (c) x < z < y (d) x > z < y
PHYSICS 1. If and Le. The correc order of % error in (a) (b) x = y > z x < z < y x > z < y. A hollow verical cylinder of radius r and heigh h has a smooh inernal surface. A small paricle is placed in conac
More informationChapter 7 Response of First-order RL and RC Circuits
Chaper 7 Response of Firs-order RL and RC Circuis 7.- The Naural Response of RL and RC Circuis 7.3 The Sep Response of RL and RC Circuis 7.4 A General Soluion for Sep and Naural Responses 7.5 Sequenial
More informationCHAPTER 12 DIRECT CURRENT CIRCUITS
CHAPTER 12 DIRECT CURRENT CIUITS DIRECT CURRENT CIUITS 257 12.1 RESISTORS IN SERIES AND IN PARALLEL When wo resisors are conneced ogeher as shown in Figure 12.1 we said ha hey are conneced in series. As
More informationd 1 = c 1 b 2 - b 1 c 2 d 2 = c 1 b 3 - b 1 c 3
and d = c b - b c c d = c b - b c c This process is coninued unil he nh row has been compleed. The complee array of coefficiens is riangular. Noe ha in developing he array an enire row may be divided or
More informationIntermediate Differential Equations Review and Basic Ideas
Inermediae Differenial Equaions Review and Basic Ideas John A. Burns Cener for Opimal Design And Conrol Inerdisciplinary Cener forappliedmahemaics Virginia Polyechnic Insiue and Sae Universiy Blacksburg,
More informationINDEX. Transient analysis 1 Initial Conditions 1
INDEX Secion Page Transien analysis 1 Iniial Condiions 1 Please inform me of your opinion of he relaive emphasis of he review maerial by simply making commens on his page and sending i o me a: Frank Mera
More informationThis exam is formed of 4 obligatory exercises in four pages numbered from 1 to 4 The use of non-programmable calculators is allowed
وزارةالتربیةوالتعلیمالعالي المدیریةالعامةللتربیة داي رةالامتحانات امتحاناتشھادةالثانویةالعامة فرع العلومالعامة مسابقةفي ال فیزیاء المدة:ثلاثساعات دورةسنة الاسم : الرقم : 005 ا لعادیة This exam is formed
More informationLab #2: Kinematics in 1-Dimension
Reading Assignmen: Chaper 2, Secions 2-1 hrough 2-8 Lab #2: Kinemaics in 1-Dimension Inroducion: The sudy of moion is broken ino wo main areas of sudy kinemaics and dynamics. Kinemaics is he descripion
More informationA First Course on Kinetics and Reaction Engineering. Class 19 on Unit 18
A Firs ourse on Kineics and Reacion Engineering lass 19 on Uni 18 Par I - hemical Reacions Par II - hemical Reacion Kineics Where We re Going Par III - hemical Reacion Engineering A. Ideal Reacors B. Perfecly
More informationKinematics Vocabulary. Kinematics and One Dimensional Motion. Position. Coordinate System in One Dimension. Kinema means movement 8.
Kinemaics Vocabulary Kinemaics and One Dimensional Moion 8.1 WD1 Kinema means movemen Mahemaical descripion of moion Posiion Time Inerval Displacemen Velociy; absolue value: speed Acceleraion Averages
More informationThis document was generated at 1:04 PM, 09/10/13 Copyright 2013 Richard T. Woodward. 4. End points and transversality conditions AGEC
his documen was generaed a 1:4 PM, 9/1/13 Copyrigh 213 Richard. Woodward 4. End poins and ransversaliy condiions AGEC 637-213 F z d Recall from Lecure 3 ha a ypical opimal conrol problem is o maimize (,,
More informationSecond Law. first draft 9/23/04, second Sept Oct 2005 minor changes 2006, used spell check, expanded example
Second Law firs draf 9/3/4, second Sep Oc 5 minor changes 6, used spell check, expanded example Kelvin-Planck: I is impossible o consruc a device ha will operae in a cycle and produce no effec oher han
More informationChapter 7: Solving Trig Equations
Haberman MTH Secion I: The Trigonomeric Funcions Chaper 7: Solving Trig Equaions Le s sar by solving a couple of equaions ha involve he sine funcion EXAMPLE a: Solve he equaion sin( ) The inverse funcions
More informationAir Traffic Forecast Empirical Research Based on the MCMC Method
Compuer and Informaion Science; Vol. 5, No. 5; 0 ISSN 93-8989 E-ISSN 93-8997 Published by Canadian Cener of Science and Educaion Air Traffic Forecas Empirical Research Based on he MCMC Mehod Jian-bo Wang,
More informationT L. t=1. Proof of Lemma 1. Using the marginal cost accounting in Equation(4) and standard arguments. t )+Π RB. t )+K 1(Q RB
Elecronic Companion EC.1. Proofs of Technical Lemmas and Theorems LEMMA 1. Le C(RB) be he oal cos incurred by he RB policy. Then we have, T L E[C(RB)] 3 E[Z RB ]. (EC.1) Proof of Lemma 1. Using he marginal
More informationSliding Mode Controller for Unstable Systems
S. SIVARAMAKRISHNAN e al., Sliding Mode Conroller for Unsable Sysems, Chem. Biochem. Eng. Q. 22 (1) 41 47 (28) 41 Sliding Mode Conroller for Unsable Sysems S. Sivaramakrishnan, A. K. Tangirala, and M.
More informationPhysics 180A Fall 2008 Test points. Provide the best answer to the following questions and problems. Watch your sig figs.
Physics 180A Fall 2008 Tes 1-120 poins Name Provide he bes answer o he following quesions and problems. Wach your sig figs. 1) The number of meaningful digis in a number is called he number of. When numbers
More information- If one knows that a magnetic field has a symmetry, one may calculate the magnitude of B by use of Ampere s law: The integral of scalar product
11.1 APPCATON OF AMPEE S AW N SYMMETC MAGNETC FEDS - f one knows ha a magneic field has a symmery, one may calculae he magniude of by use of Ampere s law: The inegral of scalar produc Closed _ pah * d
More information( ) = Q 0. ( ) R = R dq. ( t) = I t
ircuis onceps The addiion of a simple capacior o a circui of resisors allows wo relaed phenomena o occur The observaion ha he ime-dependence of a complex waveform is alered by he circui is referred o as
More informationCHAPTER 2 Signals And Spectra
CHAPER Signals And Specra Properies of Signals and Noise In communicaion sysems he received waveform is usually caegorized ino he desired par conaining he informaion, and he undesired par. he desired par
More informationElectrical Circuits. 1. Circuit Laws. Tools Used in Lab 13 Series Circuits Damped Vibrations: Energy Van der Pol Circuit
V() R L C 513 Elecrical Circuis Tools Used in Lab 13 Series Circuis Damped Vibraions: Energy Van der Pol Circui A series circui wih an inducor, resisor, and capacior can be represened by Lq + Rq + 1, a
More informationMath 116 Second Midterm March 21, 2016
Mah 6 Second Miderm March, 06 UMID: EXAM SOLUTIONS Iniials: Insrucor: Secion:. Do no open his exam unil you are old o do so.. Do no wrie your name anywhere on his exam. 3. This exam has pages including
More information5.1 - Logarithms and Their Properties
Chaper 5 Logarihmic Funcions 5.1 - Logarihms and Their Properies Suppose ha a populaion grows according o he formula P 10, where P is he colony size a ime, in hours. When will he populaion be 2500? We
More informationElectrical and current self-induction
Elecrical and curren self-inducion F. F. Mende hp://fmnauka.narod.ru/works.hml mende_fedor@mail.ru Absrac The aricle considers he self-inducance of reacive elemens. Elecrical self-inducion To he laws of
More informationUnit 1 Test Review Physics Basics, Movement, and Vectors Chapters 1-3
A.P. Physics B Uni 1 Tes Reiew Physics Basics, Moemen, and Vecors Chapers 1-3 * In sudying for your es, make sure o sudy his reiew shee along wih your quizzes and homework assignmens. Muliple Choice Reiew:
More informationODEs II, Lecture 1: Homogeneous Linear Systems - I. Mike Raugh 1. March 8, 2004
ODEs II, Lecure : Homogeneous Linear Sysems - I Mike Raugh March 8, 4 Inroducion. In he firs lecure we discussed a sysem of linear ODEs for modeling he excreion of lead from he human body, saw how o ransform
More information15210 RECORDING TIMER - AC STUDENT NAME:
CONTENTS 15210 RECORDING TIMER - AC STUDENT NAME: REQUIRED ACCESSORIES o C-Clamps o Ruler or Meer Sick o Mass (200 g or larger) OPTIONAL ACCESSORIES o Ticker Tape Dispenser (#15225) o Consan Speed Vehicle
More informationLecture 20: Riccati Equations and Least Squares Feedback Control
34-5 LINEAR SYSTEMS Lecure : Riccai Equaions and Leas Squares Feedback Conrol 5.6.4 Sae Feedback via Riccai Equaions A recursive approach in generaing he marix-valued funcion W ( ) equaion for i for he
More information3.1.3 INTRODUCTION TO DYNAMIC OPTIMIZATION: DISCRETE TIME PROBLEMS. A. The Hamiltonian and First-Order Conditions in a Finite Time Horizon
3..3 INRODUCION O DYNAMIC OPIMIZAION: DISCREE IME PROBLEMS A. he Hamilonian and Firs-Order Condiions in a Finie ime Horizon Define a new funcion, he Hamilonian funcion, H. H he change in he oal value of
More informationSINUSOIDAL WAVEFORMS
SINUSOIDAL WAVEFORMS The sinusoidal waveform is he only waveform whose shape is no affeced by he response characerisics of R, L, and C elemens. Enzo Paerno CIRCUIT ELEMENTS R [ Ω ] Resisance: Ω: Ohms Georg
More informationRobotics I. April 11, The kinematics of a 3R spatial robot is specified by the Denavit-Hartenberg parameters in Tab. 1.
Roboics I April 11, 017 Exercise 1 he kinemaics of a 3R spaial robo is specified by he Denavi-Harenberg parameers in ab 1 i α i d i a i θ i 1 π/ L 1 0 1 0 0 L 3 0 0 L 3 3 able 1: able of DH parameers of
More informationMultiple Choice Solutions 1. E (2003 AB25) () xt t t t 2. A (2008 AB21/BC21) 3. B (2008 AB7) Using Fundamental Theorem of Calculus: 1
Paricle Moion Soluions We have inenionally included more maerial han can be covered in mos Suden Sudy Sessions o accoun for groups ha are able o answer he quesions a a faser rae. Use your own judgmen,
More information