Two Dimensional Numerical Simulator for Modeling NDC Region in SNDC Devices

Size: px
Start display at page:

Download "Two Dimensional Numerical Simulator for Modeling NDC Region in SNDC Devices"

Transcription

1 Journal of Physics: Conference Series PAPER OPEN ACCESS Two Dimensional Numerical Simulator for Moeling NDC Region in SNDC Devices To cite this article: Dheeraj Kumar Sinha et al 2016 J. Phys.: Conf. Ser View the article online for upates an enhancements. Relate content - Self-sustaine oscillations in weakly couple GaAs/AlGaAs superlattices G K Rasulova, M V Golubkov, A V Leonov et al. - Electron transport an ionization coefficients in C2F6 an its mixtures J e Urquijo, A A Castrejón-Pita, J L Hernánez-Ávila et al. - Operation of resonant-tunnelling-ioe oscillators beyon tunnel-lifetime limit at 564 GHz M. Feiginov, C. Sylo, O. Cojocari et al. This content was ownloae from IP aress on 13/09/2018 at 02:58

2 oi: / /759/1/ Two Dimensional Numerical Simulator for Moeling NDC Region in SNDC Devices Dheeraj Kumar Sinha, Amitabh Chatterjee an Gaurav Trivei Department of Electronics an Electrical Engineering, Inian Institute of Technology Guwahati, Assam, Inia Abstract. This paper eals about making a numerical simulator for observing the self-organization phenomenon like current filament formation in the negative ifferential conuctivity (NDC) region of operation in a evice using MATLAB environment. Emphasis is lai on choosing the appropriate bounary conitions in simulator to observe the NDC part of the evice Current-Voltage (I V ) characteristics along with the appropriate isothermal physical moels are inclue for mobility an the recombination-generation. The simulator use in this paper is esigne to observe NDC region in S-shape NDC (SNDC) evices like BJT an similar conitions can be use to observe the fiel omains in the N-shape NDC (NNDC) evices with appropriate changes. Dirichlet an Neumann bounary conitions are applie to BJT an I V characteristics of the evelope evice simulator are compare with the commercial available TCAD evice simulator (i.e., Sentaurus). 1. Introuction Detaile numerical simulation an moeling have become extremely important over the last four ecaes to optimize the performance an esign of nano-scale semiconuctor evices. It has been seen that huge number of trials are require in the experiments to optimize semiconuctor evice performance however it can be ecimate using computer simulation for faster evelopment cycle 1. Numerical moeling of semiconuctor evices are base on basic semiconuctor equations which has been first propose by Gummel et. al in 1964 an many papers have been publishe in this omain 2. In general, the numerical simulation of the evice inclues a basic semiconuctor physical moels which escribes carrier transport moels in material such as: rift-iffusion, hyroynamic etc3. Moreover, we solve Poisson s an continuity equations for both electron an hole, to more computationally challenging physical moels such as the energy balance, which exemplifies some higher moment simplification of Boltzmann transport approximation (BTE). However, the choice of appropriate physical moels are epenent on level of etail is require. In particular, semiconuctor evice structures whose electronic properties changes on a eep nanometer scale gives an number of examples of nonlinear transport moels4, 5. In these evices nonlinear transport mechanisms are complex because electrical or optical properties are governe by ifferent nonlinear ynamic processes. The electronic transport properties of a semiconuctor material show up most by I V characteristic form uner time-inepenent voltage bias conitions4. It has been etermine in a complex manner by microscopic properties of material, which specifies the current ensity (J) with function of local electric fiel (E) an by electroes. Uner low voltage or current conitions, the relation between J an E o not always exist, however it is strongly epenent in high voltage or current conitions. Moreover, Content from this work may be use uner the terms of the Creative Commons Attribution 3.0 licence. Any further istribution of this work must maintain attribution to the author(s) an the title of the work, journal citation an DOI. Publishe uner licence by Lt 1

3 oi: / /759/1/ Current (A). Current (A) N shape Negative ifferential resistance S shape Negative ifferential resistance Voltage (V) Voltage (V). Figure 1. Typical negative ifferential conuctivity (NDC) regions are observe in I V characteristics of the evice structure uner high current or voltage conitions6. an unstable situation has been observe when current ensity increases or ecreases correspons to negative ifferential conuctivity region6. Two ifferent I V characteristics of negative ifferential conuctivity (NDC) regions have been escribe by S-shape or N-shape J(E) characteristic, which have been enote by SNDC an NNDC respectively, as shown in Figure 1. The NDC nature of the I V characteristics have been analyze analytically an the conitions uner which SNDC an NNDC curves are forme have been iscusse by Scholl et al.6. The physics behin the formation of current filamentation in SNDC evices an Fiel omains in NNDC evices have also been explaine6. There has been research work one on how to make a evice simulator by solving the semiconuctor equations an about the NDC operating region of the evice iniviually but very little emphasis has been lai over the conitions to be satisfie by a simulator to observe the NDC region of the I V characteristics of a evice. This paper targets at making a simulator to observe the NDC nature of the SNDC like BJT by applying appropriate bounary conitions. In this paper a generalize metho is propose to apply the appropriate bounary conitions to observe the NDC region of the I V characteristics an the same is prove by plotting the SNDC I V characteristics of a BJT. Both Neumann an Dirichlet bounary conitions are applie to the BJT an a comparison is mae on choosing ifferent bounary conitions. The paper is organize as follows: Section II briefs flow of the propose simulator. Furthermore, two-imensional iscretization scheme is formulate in terms of basic semiconuctor equations then the bounary conitions for observing the NDC curves is iscusse. Finally, the results are presente an iscusse for ifferent bounary conitions. 2. Propose Flow of Device Simulator The basic flow of the simulator an evice structure use for simulation purpose are shown in Figure 2. The structure for which the simulator is built an bounary conitions teste is a two imensional (2D) NPN BJT, as shown in Figure 2(b). The structure has three contacts base, emitter an collector out of which emitter an collector contacts are heavily ope n + regions with N D = cm 3 an base contact is heavily ope p + region with N A = cm 3. The collector-base junction is forme by relatively less ope collector an base regions with N Dn = cm 3 an N Ap = cm 3 respectively. The contact regions nee to be heavily ope to have better contacts an lightly ope region near the junction helps in withstaning the practical levels of operating voltage not limite by tunneling effect. The propose flow of evice simulator is escribe below: (i) Using the bounary conitions as input an the matrix representation of iscretize 2

4 oi: / /759/1/ Doping Concentration Mobility an Generation Recombination moels Poisson s Equation solver Continuity equation for electrons Continuity equation for holes Input Voltage/Current as Dirichlet/Neumann BC 0.5 u m 0.5 u m 0.25 u m Emitter Contact Base Contact N Emitter Na = N=1e20 1e19 P + N+ 0.3 u m 1 u m Convergent No P Base Na = 1e17 Poisson s equation solver N Collector N = 1e16 3 u m 0.5 u m Convergent No Increment the input Carrier transport equation Solving for Quasi steay state No No Final value of input Voltage reache N = 1e20 N u m Plot the Voltage an current istribution Plot I-V characteristics En of Simulation En of Simulation Collector Contact 1 u m (a) (b) Figure 2. (a) Propose flow of the evice simulator, (b) Device uner test for simulation purpose is layere BJT evice structure where oping of all the regions are given. Poisson s equation, etermines the potential at every iscretize noe in the evice structure. (ii) From the potential obtaine in Step 1, we calculate the Electric fiel at every point using central ifference FDM metho. (iii) It can be seen that, as the mobility an net recombination rate are inclue into the continuity equations results non linear partial ifferential equation which cannot be solve irectly by matrix metho. Therefore, to solve these equations, we use Gummels iteration metho 2, while solving the electron continuity equation consier hole concentration (p) as constant (equal to initial oping concentration for the first time) an calculate electron concentration (n). (iv) While solving the hole continuity equation consier electron concentration as a constant an equal to that obtaine in step 3 an calculate the hole concentration. Now upate the hole concentration, mobility an recombination rate in step 3 an repeat these two steps till convergence is reache. At this point the electron an hole concentrations are obtaine will be satisfying both the continuity equations. (v) Substitute the electron an hole concentration obtaine after convergence in step 4 into the Poissons equation an solve step 2 to step 5 until convergence is achieve. At this point the potential an carrier concentrations obtaine will satisfy the Poisson s an continuity equations. Now we calculate the current ensity using the Drift-Diffusion equations. 3

5 oi: / /759/1/ (vi) Finally we plot the I-V characteristics by increasing the input an repeat step 1 to step 5 until the final value of input is reache Basic Semiconuctor Equations To analyze the istribution of current ensity an potential insie any arbitrary semiconuctor evice uner various operating conitions it nees to be moele mathematically using the following equations: (i) Poisson s equation (ii) Continuity equations for electrons an holes (iii) Carrier transport equation for electrons an holes (iv) Mobility an Recombination-Generation rate 2.2. Discretization of Basic semiconuctor equations We nee to iscretize the basic semiconuctor equations to solve them numerically. Though there are many ways of iscretization of the Partial ifferential equations like finite element metho (FEM), finite volume metho (FVM) an finite ifference metho (FDM). However, we choose to iscretize using five point Finite Difference Metho owing to its simplicity an goo accuracy. FDM Discretize basic semiconuctor equations are: Poisson s equation: n ψ(x + 1, y) + n ( n ψ(x 1, y) 2 + m ) + m ψ(x, y + 1) + m ψ(x, y 1) = m m m n n n q m.n ϵ (n p C) (1) Continuity equation for electrons: n n(x + 1, y) n E x + n(x 1, y) m n(x, y 1) n m + n m n m E y E x n(x, y) m + n(x, y + 1) m E y + n ( m 2 + n ) + temp = n m ( R(ψ, n, p) m n + 1 ) n D n D n t (2) Continuity equation for holes: n p(x + 1, y) + n E x + p(x 1, y) m p(x, y 1) n m n m n m E y E x p(x, y) m + p(x, y + 1) 2 ( m n + n m m n ( R(ψ, n, p) D p + m E y + n ) temp = + 1 ) p D p t (3) Where E x = ψ x ; E x = 2 ψ x 2 ; an E y = ψ y ; E y = 2 ψ y 2 ; m /n = horizontal/vertical ifferential length an E x, E x, E y an E y in the continuity equation can be calculate using the central ifference an five point iscretization metho. 4

6 XXVII IUPAP Conference on Computational Physics (CCP2015) oi: / /759/1/ x 10 3 BJT I V characteristics with Dirichlet Bounary Conitions V be = 1 V BJT I V characteristics with Neumann Bounary Conitions V be = 1 V I c in A 4 I c in A V ce in V V ce in V (a) (b) Figure 3. Propose evice simulator results when ifferent bounary conitions are consiere. (a) Dirichlet bounary conitions an voltage input are consiere, (b) Neumann bounary conitions an current input are chosen Bounary Conitions For Observing NDC Region Bounaries of a semiconuctor evice can be classifie into two types such as: Physical Bounaries an Artificial Bounaries. Physical bounaries inclue the bounaries which physically exist like contacts an interface to the insulating material while artificial bounaries are the ones which is introuce for the purpose of simulation or calculations so that it can separate one evice from the neighboring evices. Bounary conitions are mathematical conitions which are specifie at the bounaries to solve the partial ifferential equations. In general the bounary conitions are ecie by the type of contacts which is epenent on the type of the input given to the evice. Broaly bounary conitions are ivie as: Dirichlet bounary: Dirichlet bounary conition states the value of the solution parameter so that solution of the partial ifferential equation nees to take at the bounaries of the omain. We solve the Poissons equation with potential as the parameter an stating Dirichlet bounary conition at the bounaries which means we are fixing the value of the potential at the bounaries which is equivalent to connecting the evice to a voltage source. To plot the I-V characteristics of a evice with Dirichlet bounary conitions is physically equivalent to connecting the evice to a variable voltage source an checking the current flowing through the evice. Neumann bounary: Neumann bounary conition states the erivative of the of the solution parameter on the bounary of the omain that must be satisfie by the solution of the partial ifferential equation. Physically Neumann bounary conition is applie when you have to connect the evice to a current source. When a current source is to be connecte to the evice it can be one in simulation terms by applying Neumann bounary conitions. 3. Results an Discussion In the simulation it is observe that, when we apply Dirichlet bounary conitions to an SNDC evice like BJT, the resulting I V characteristics are negative ifferential resistance as shown in Figure 3(a). But in the most cases when we use Newtons metho, it oesn t converge at the first breakown point from where the curve goes into NDC part. This is ue to triggering of impact ionization in the evice, which results avalanche-generate electrons an avalanche-generate holes thus current sharply increases across the terminal which results in instability in the evice an oes not satisfy the outer bounary conitions. However, applying Neumann bounary conition at collector contact of the BJT an Dirichlet bounary conitions at emitter an base contact, shows S-shape negative ifferential resistance region as shown in Figure 3(b). 5

7 oi: / /759/1/ Comparision between TCAD an Matlab Simulation result V be = 1 V 10 1 I c in A MATLAB RESULT TCAD simulate result V ce in V Figure 4. Comparison between commercial evice simulator (Sentaurus) results an our propose evice simulator result of BJT I V characteristics with base voltage = 1 V an Neumann Bounary conitions. To verify our results with commercial evice simulator we have use Sentaurus, which is a set of TCAD tools which simulates an characterize semiconuctor evices. The BJT given in Figure 4 is compare simulate results with the same base voltage an bounary conitions but using Sentaurus TCAD an the result obtaine is exporte an plotte in matlab an compare with the result observe from matlab. The output is showing some variation between the matlab simulate graph an TCAD simulate graph this might be because of reasons like ifferent mobility moels being use an the effects like temperature which is not use in the matlab simulator but are taken into consieration in sentaurus TCAD. 4. Conclusion In summary, we have evelope a numerical evice simulator to observe negative ifferential conuctivity region in SNDC evices such as BJT. For the simulation purpose, we have use BJT as an evice structure an applie ifferent bounary conitions across the collector terminal, It has been observe that Dirichlet bounary conitions shows NNDC behavior while applying Neumann bounary conition results SNDC nature of the evice. The propose flow of the evice simulator has been iscusse an further results of evelope evice simulator have been compare with the commercial TCAD evice simulator (i.e, Sentaurus). References 1 P. Dubock, DC Numerical Moel for Arbitrarily biase Bipolar Transistors in Two Dimensions, Electronics Letters, vol. 6, no. 3, pp , Feb H. Gummel, A Self-consistent Iterative Scheme for One-imensional Steay-state Transistor Calculations, IEEE Transactions on Electron Devices, vol. 11, no. 10, pp , Oct J. Slotboom, Iterative Scheme for one an Two imensional DC Transistor Simulation, Electronics Letters, vol. 5, no. 26, pp , Dec D. Kenney an R. OBrien, Two-imensional Mathematical Analysis of a Planar type Junction Fiel-Effect Transistor, IBM Journal of Research an Development, vol. 13, no. 6, pp , Nov S. Selberherr, Analysis an Simulation of Semiconuctor Devices, 1st e. Springer Science an Business Meia, E. Scholl, Equal Areas Rules for Filamentation in SNDC Elements, Soli-State Electronics, vol. 29, no. 7, pp ,

EE 330 Lecture 12. Devices in Semiconductor Processes. Diodes

EE 330 Lecture 12. Devices in Semiconductor Processes. Diodes EE 330 Lecture 12 evices in Semiconuctor Processes ioes Review from Last Lecture http://www.ayah.com/perioic/mages/perioic%20table.png Review from Last Lecture Review from Last Lecture Silicon opants in

More information

Lecture contents. Metal-semiconductor contact

Lecture contents. Metal-semiconductor contact 1 Lecture contents Metal-semiconuctor contact Electrostatics: Full epletion approimation Electrostatics: Eact electrostatic solution Current Methos for barrier measurement Junctions: general approaches,

More information

A Novel Decoupled Iterative Method for Deep-Submicron MOSFET RF Circuit Simulation

A Novel Decoupled Iterative Method for Deep-Submicron MOSFET RF Circuit Simulation A Novel ecouple Iterative Metho for eep-submicron MOSFET RF Circuit Simulation CHUAN-SHENG WANG an YIMING LI epartment of Mathematics, National Tsing Hua University, National Nano evice Laboratories, an

More information

An inductance lookup table application for analysis of reluctance stepper motor model

An inductance lookup table application for analysis of reluctance stepper motor model ARCHIVES OF ELECTRICAL ENGINEERING VOL. 60(), pp. 5- (0) DOI 0.478/ v07-0-000-y An inuctance lookup table application for analysis of reluctance stepper motor moel JAKUB BERNAT, JAKUB KOŁOTA, SŁAWOMIR

More information

05 The Continuum Limit and the Wave Equation

05 The Continuum Limit and the Wave Equation Utah State University DigitalCommons@USU Founations of Wave Phenomena Physics, Department of 1-1-2004 05 The Continuum Limit an the Wave Equation Charles G. Torre Department of Physics, Utah State University,

More information

Dusty Plasma Void Dynamics in Unmoving and Moving Flows

Dusty Plasma Void Dynamics in Unmoving and Moving Flows 7 TH EUROPEAN CONFERENCE FOR AERONAUTICS AND SPACE SCIENCES (EUCASS) Dusty Plasma Voi Dynamics in Unmoving an Moving Flows O.V. Kravchenko*, O.A. Azarova**, an T.A. Lapushkina*** *Scientific an Technological

More information

EE 330 Lecture 13. Devices in Semiconductor Processes. Diodes Capacitors Transistors

EE 330 Lecture 13. Devices in Semiconductor Processes. Diodes Capacitors Transistors EE 330 Lecture 13 evices in Semiconuctor Processes ioes Capacitors Transistors Review from Last Lecture pn Junctions Physical Bounary Separating n-type an p-type regions Extens farther into n-type region

More information

Chapter 9 Method of Weighted Residuals

Chapter 9 Method of Weighted Residuals Chapter 9 Metho of Weighte Resiuals 9- Introuction Metho of Weighte Resiuals (MWR) is an approimate technique for solving bounary value problems. It utilizes a trial functions satisfying the prescribe

More information

Revisiting the Charge Concept in HBT/BJT Models

Revisiting the Charge Concept in HBT/BJT Models evisiting the Charge Concept in HBT/BJT Moels Zoltan Huska an Ehrenfrie Seebacher austriamicrosystems AG 23r Bipolar Arbeitkeis BipAK Meeting at STM Crolles, France, 5 October 2 Outline recalling the junction

More information

3-D FEM Modeling of fiber/matrix interface debonding in UD composites including surface effects

3-D FEM Modeling of fiber/matrix interface debonding in UD composites including surface effects IOP Conference Series: Materials Science an Engineering 3-D FEM Moeling of fiber/matrix interface eboning in UD composites incluing surface effects To cite this article: A Pupurs an J Varna 2012 IOP Conf.

More information

12.11 Laplace s Equation in Cylindrical and

12.11 Laplace s Equation in Cylindrical and SEC. 2. Laplace s Equation in Cylinrical an Spherical Coorinates. Potential 593 2. Laplace s Equation in Cylinrical an Spherical Coorinates. Potential One of the most important PDEs in physics an engineering

More information

Thermal conductivity of graded composites: Numerical simulations and an effective medium approximation

Thermal conductivity of graded composites: Numerical simulations and an effective medium approximation JOURNAL OF MATERIALS SCIENCE 34 (999)5497 5503 Thermal conuctivity of grae composites: Numerical simulations an an effective meium approximation P. M. HUI Department of Physics, The Chinese University

More information

ECE 422 Power System Operations & Planning 7 Transient Stability

ECE 422 Power System Operations & Planning 7 Transient Stability ECE 4 Power System Operations & Planning 7 Transient Stability Spring 5 Instructor: Kai Sun References Saaat s Chapter.5 ~. EPRI Tutorial s Chapter 7 Kunur s Chapter 3 Transient Stability The ability of

More information

Switching Time Optimization in Discretized Hybrid Dynamical Systems

Switching Time Optimization in Discretized Hybrid Dynamical Systems Switching Time Optimization in Discretize Hybri Dynamical Systems Kathrin Flaßkamp, To Murphey, an Sina Ober-Blöbaum Abstract Switching time optimization (STO) arises in systems that have a finite set

More information

inflow outflow Part I. Regular tasks for MAE598/494 Task 1

inflow outflow Part I. Regular tasks for MAE598/494 Task 1 MAE 494/598, Fall 2016 Project #1 (Regular tasks = 20 points) Har copy of report is ue at the start of class on the ue ate. The rules on collaboration will be release separately. Please always follow the

More information

EE 330 Lecture 15. Devices in Semiconductor Processes. Diodes Capacitors MOSFETs

EE 330 Lecture 15. Devices in Semiconductor Processes. Diodes Capacitors MOSFETs EE 330 Lecture 15 evices in Semiconuctor Processes ioes Capacitors MOSFETs Review from Last Lecture Basic evices an evice Moels Resistor ioe Capacitor MOSFET BJT Review from Last Lecture Review from Last

More information

On the number of isolated eigenvalues of a pair of particles in a quantum wire

On the number of isolated eigenvalues of a pair of particles in a quantum wire On the number of isolate eigenvalues of a pair of particles in a quantum wire arxiv:1812.11804v1 [math-ph] 31 Dec 2018 Joachim Kerner 1 Department of Mathematics an Computer Science FernUniversität in

More information

A Simple Model for the Calculation of Plasma Impedance in Atmospheric Radio Frequency Discharges

A Simple Model for the Calculation of Plasma Impedance in Atmospheric Radio Frequency Discharges Plasma Science an Technology, Vol.16, No.1, Oct. 214 A Simple Moel for the Calculation of Plasma Impeance in Atmospheric Raio Frequency Discharges GE Lei ( ) an ZHANG Yuantao ( ) Shanong Provincial Key

More information

Module 2. DC Circuit. Version 2 EE IIT, Kharagpur

Module 2. DC Circuit. Version 2 EE IIT, Kharagpur Moule 2 DC Circuit Lesson 9 Analysis of c resistive network in presence of one non-linear element Objectives To unerstan the volt (V ) ampere ( A ) characteristics of linear an nonlinear elements. Concept

More information

Magnetic field generated by current filaments

Magnetic field generated by current filaments Journal of Phsics: Conference Series OPEN ACCESS Magnetic fiel generate b current filaments To cite this article: Y Kimura 2014 J. Phs.: Conf. Ser. 544 012004 View the article online for upates an enhancements.

More information

AIEEE Physics Model Question Paper

AIEEE Physics Model Question Paper IEEE Physics Moel Question Paper ote: Question o. 11 to 1 an 1 to consist of Eight (8) marks each for each correct response an remaining questions consist of Four (4) marks. ¼ marks will be eucte for inicating

More information

Sensors & Transducers 2015 by IFSA Publishing, S. L.

Sensors & Transducers 2015 by IFSA Publishing, S. L. Sensors & Transucers, Vol. 184, Issue 1, January 15, pp. 53-59 Sensors & Transucers 15 by IFSA Publishing, S. L. http://www.sensorsportal.com Non-invasive an Locally Resolve Measurement of Soun Velocity

More information

Approaches for Predicting Collection Efficiency of Fibrous Filters

Approaches for Predicting Collection Efficiency of Fibrous Filters Volume 5, Issue, Summer006 Approaches for Preicting Collection Efficiency of Fibrous Filters Q. Wang, B. Maze, H. Vahei Tafreshi, an B. Poureyhimi Nonwovens Cooperative esearch Center, North Carolina State

More information

Patterns in bistable resonant-tunneling structures

Patterns in bistable resonant-tunneling structures PHYSICAL REVIEW B VOLUME 56, NUMBER 20 Patterns in bistable resonant-tunneling structures 15 NOVEMBER 1997-II B. A. Glavin an V. A. Kochelap Institute of Semiconuctor Physics, Ukrainian Acaemy of Sciences,

More information

4. Important theorems in quantum mechanics

4. Important theorems in quantum mechanics TFY4215 Kjemisk fysikk og kvantemekanikk - Tillegg 4 1 TILLEGG 4 4. Important theorems in quantum mechanics Before attacking three-imensional potentials in the next chapter, we shall in chapter 4 of this

More information

APPROXIMATE SOLUTION FOR TRANSIENT HEAT TRANSFER IN STATIC TURBULENT HE II. B. Baudouy. CEA/Saclay, DSM/DAPNIA/STCM Gif-sur-Yvette Cedex, France

APPROXIMATE SOLUTION FOR TRANSIENT HEAT TRANSFER IN STATIC TURBULENT HE II. B. Baudouy. CEA/Saclay, DSM/DAPNIA/STCM Gif-sur-Yvette Cedex, France APPROXIMAE SOLUION FOR RANSIEN HEA RANSFER IN SAIC URBULEN HE II B. Bauouy CEA/Saclay, DSM/DAPNIA/SCM 91191 Gif-sur-Yvette Ceex, France ABSRAC Analytical solution in one imension of the heat iffusion equation

More information

Schrödinger s equation.

Schrödinger s equation. Physics 342 Lecture 5 Schröinger s Equation Lecture 5 Physics 342 Quantum Mechanics I Wenesay, February 3r, 2010 Toay we iscuss Schröinger s equation an show that it supports the basic interpretation of

More information

Model for Dopant and Impurity Segregation During Vapor Phase Growth

Model for Dopant and Impurity Segregation During Vapor Phase Growth Mat. Res. Soc. Symp. Proc. Vol. 648, P3.11.1-7 2001 Materials Research Society Moel for Dopant an Impurity Segregation During Vapor Phase Growth Craig B. Arnol an Michael J. Aziz Division of Engineering

More information

Conservation laws a simple application to the telegraph equation

Conservation laws a simple application to the telegraph equation J Comput Electron 2008 7: 47 51 DOI 10.1007/s10825-008-0250-2 Conservation laws a simple application to the telegraph equation Uwe Norbrock Reinhol Kienzler Publishe online: 1 May 2008 Springer Scienceusiness

More information

Introduction to the Vlasov-Poisson system

Introduction to the Vlasov-Poisson system Introuction to the Vlasov-Poisson system Simone Calogero 1 The Vlasov equation Consier a particle with mass m > 0. Let x(t) R 3 enote the position of the particle at time t R an v(t) = ẋ(t) = x(t)/t its

More information

Planar sheath and presheath

Planar sheath and presheath 5/11/1 Flui-Poisson System Planar sheath an presheath 1 Planar sheath an presheath A plasma between plane parallel walls evelops a positive potential which equalizes the rate of loss of electrons an ions.

More information

Calculus Class Notes for the Combined Calculus and Physics Course Semester I

Calculus Class Notes for the Combined Calculus and Physics Course Semester I Calculus Class Notes for the Combine Calculus an Physics Course Semester I Kelly Black December 14, 2001 Support provie by the National Science Founation - NSF-DUE-9752485 1 Section 0 2 Contents 1 Average

More information

Chapter 4. Electrostatics of Macroscopic Media

Chapter 4. Electrostatics of Macroscopic Media Chapter 4. Electrostatics of Macroscopic Meia 4.1 Multipole Expansion Approximate potentials at large istances 3 x' x' (x') x x' x x Fig 4.1 We consier the potential in the far-fiel region (see Fig. 4.1

More information

A Course in Machine Learning

A Course in Machine Learning A Course in Machine Learning Hal Daumé III 12 EFFICIENT LEARNING So far, our focus has been on moels of learning an basic algorithms for those moels. We have not place much emphasis on how to learn quickly.

More information

Harmonic Modelling of Thyristor Bridges using a Simplified Time Domain Method

Harmonic Modelling of Thyristor Bridges using a Simplified Time Domain Method 1 Harmonic Moelling of Thyristor Briges using a Simplifie Time Domain Metho P. W. Lehn, Senior Member IEEE, an G. Ebner Abstract The paper presents time omain methos for harmonic analysis of a 6-pulse

More information

SiC-based Power Converters for High Temperature Applications

SiC-based Power Converters for High Temperature Applications Materials Science orum Vols. 556-557 (7) pp 965-97 online at http://www.scientific.net (7) Trans Tech Publications Switzerlan Online available since 7/Sep/5 -base Power Converters for High Temperature

More information

arxiv: v2 [cond-mat.stat-mech] 11 Nov 2016

arxiv: v2 [cond-mat.stat-mech] 11 Nov 2016 Noname manuscript No. (will be inserte by the eitor) Scaling properties of the number of ranom sequential asorption iterations neee to generate saturate ranom packing arxiv:607.06668v2 [con-mat.stat-mech]

More information

Separation of Variables

Separation of Variables Physics 342 Lecture 1 Separation of Variables Lecture 1 Physics 342 Quantum Mechanics I Monay, January 25th, 2010 There are three basic mathematical tools we nee, an then we can begin working on the physical

More information

arxiv:physics/ v2 [physics.ed-ph] 23 Sep 2003

arxiv:physics/ v2 [physics.ed-ph] 23 Sep 2003 Mass reistribution in variable mass systems Célia A. e Sousa an Vítor H. Rorigues Departamento e Física a Universiae e Coimbra, P-3004-516 Coimbra, Portugal arxiv:physics/0211075v2 [physics.e-ph] 23 Sep

More information

THE VAN KAMPEN EXPANSION FOR LINKED DUFFING LINEAR OSCILLATORS EXCITED BY COLORED NOISE

THE VAN KAMPEN EXPANSION FOR LINKED DUFFING LINEAR OSCILLATORS EXCITED BY COLORED NOISE Journal of Soun an Vibration (1996) 191(3), 397 414 THE VAN KAMPEN EXPANSION FOR LINKED DUFFING LINEAR OSCILLATORS EXCITED BY COLORED NOISE E. M. WEINSTEIN Galaxy Scientific Corporation, 2500 English Creek

More information

24th European Photovoltaic Solar Energy Conference, September 2009, Hamburg, Germany

24th European Photovoltaic Solar Energy Conference, September 2009, Hamburg, Germany 4th European hotovoltaic Solar Energy Conference, 1-5 September 9, Hamburg, Germany LOCK-IN THERMOGRAHY ON CRYSTALLINE SILICON ON GLASS (CSG) THIN FILM MODULES: INFLUENCE OF ELTIER CONTRIBUTIONS H. Straube,

More information

Tutorial Test 5 2D welding robot

Tutorial Test 5 2D welding robot Tutorial Test 5 D weling robot Phys 70: Planar rigi boy ynamics The problem statement is appene at the en of the reference solution. June 19, 015 Begin: 10:00 am En: 11:30 am Duration: 90 min Solution.

More information

Thermal runaway during blocking

Thermal runaway during blocking Thermal runaway uring blocking CES_stable CES ICES_stable ICES k 6.5 ma 13 6. 12 5.5 11 5. 1 4.5 9 4. 8 3.5 7 3. 6 2.5 5 2. 4 1.5 3 1. 2.5 1. 6 12 18 24 3 36 s Thermal runaway uring blocking Application

More information

Static Analysis of Electrostatically Actuated Micro Cantilever Beam

Static Analysis of Electrostatically Actuated Micro Cantilever Beam Available online at www.scienceirect.com Proceia Engineering 51 ( 13 ) 776 78 Chemical, Civil an Mechanical Engineering Tracks of 3r Nirma University International Conference on Engineering (NUiCONE 1)

More information

A SIMPLE ENGINEERING MODEL FOR SPRINKLER SPRAY INTERACTION WITH FIRE PRODUCTS

A SIMPLE ENGINEERING MODEL FOR SPRINKLER SPRAY INTERACTION WITH FIRE PRODUCTS International Journal on Engineering Performance-Base Fire Coes, Volume 4, Number 3, p.95-3, A SIMPLE ENGINEERING MOEL FOR SPRINKLER SPRAY INTERACTION WITH FIRE PROCTS V. Novozhilov School of Mechanical

More information

Computing Exact Confidence Coefficients of Simultaneous Confidence Intervals for Multinomial Proportions and their Functions

Computing Exact Confidence Coefficients of Simultaneous Confidence Intervals for Multinomial Proportions and their Functions Working Paper 2013:5 Department of Statistics Computing Exact Confience Coefficients of Simultaneous Confience Intervals for Multinomial Proportions an their Functions Shaobo Jin Working Paper 2013:5

More information

Chapter 2 Lagrangian Modeling

Chapter 2 Lagrangian Modeling Chapter 2 Lagrangian Moeling The basic laws of physics are use to moel every system whether it is electrical, mechanical, hyraulic, or any other energy omain. In mechanics, Newton s laws of motion provie

More information

Application of the homotopy perturbation method to a magneto-elastico-viscous fluid along a semi-infinite plate

Application of the homotopy perturbation method to a magneto-elastico-viscous fluid along a semi-infinite plate Freun Publishing House Lt., International Journal of Nonlinear Sciences & Numerical Simulation, (9), -, 9 Application of the homotopy perturbation metho to a magneto-elastico-viscous flui along a semi-infinite

More information

Goal of this chapter is to learn what is Capacitance, its role in electronic circuit, and the role of dielectrics.

Goal of this chapter is to learn what is Capacitance, its role in electronic circuit, and the role of dielectrics. PHYS 220, Engineering Physics, Chapter 24 Capacitance an Dielectrics Instructor: TeYu Chien Department of Physics an stronomy University of Wyoming Goal of this chapter is to learn what is Capacitance,

More information

Spring 2016 Network Science

Spring 2016 Network Science Spring 206 Network Science Sample Problems for Quiz I Problem [The Application of An one-imensional Poisson Process] Suppose that the number of typographical errors in a new text is Poisson istribute with

More information

Strength Analysis of CFRP Composite Material Considering Multiple Fracture Modes

Strength Analysis of CFRP Composite Material Considering Multiple Fracture Modes 5--XXXX Strength Analysis of CFRP Composite Material Consiering Multiple Fracture Moes Author, co-author (Do NOT enter this information. It will be pulle from participant tab in MyTechZone) Affiliation

More information

Design A Robust Power System Stabilizer on SMIB Using Lyapunov Theory

Design A Robust Power System Stabilizer on SMIB Using Lyapunov Theory Design A Robust Power System Stabilizer on SMIB Using Lyapunov Theory Yin Li, Stuent Member, IEEE, Lingling Fan, Senior Member, IEEE Abstract This paper proposes a robust power system stabilizer (PSS)

More information

PCCP PAPER. 1 Introduction. A. Nenning,* A. K. Opitz, T. M. Huber and J. Fleig. View Article Online View Journal View Issue

PCCP PAPER. 1 Introduction. A. Nenning,* A. K. Opitz, T. M. Huber and J. Fleig. View Article Online View Journal View Issue PAPER View Article Online View Journal View Issue Cite this: Phys. Chem. Chem. Phys., 2014, 16, 22321 Receive 4th June 2014, Accepte 3r September 2014 DOI: 10.1039/c4cp02467b www.rsc.org/pccp 1 Introuction

More information

'HVLJQ &RQVLGHUDWLRQ LQ 0DWHULDO 6HOHFWLRQ 'HVLJQ 6HQVLWLYLW\,1752'8&7,21

'HVLJQ &RQVLGHUDWLRQ LQ 0DWHULDO 6HOHFWLRQ 'HVLJQ 6HQVLWLYLW\,1752'8&7,21 Large amping in a structural material may be either esirable or unesirable, epening on the engineering application at han. For example, amping is a esirable property to the esigner concerne with limiting

More information

Optimized Schwarz Methods with the Yin-Yang Grid for Shallow Water Equations

Optimized Schwarz Methods with the Yin-Yang Grid for Shallow Water Equations Optimize Schwarz Methos with the Yin-Yang Gri for Shallow Water Equations Abessama Qaouri Recherche en prévision numérique, Atmospheric Science an Technology Directorate, Environment Canaa, Dorval, Québec,

More information

Closed and Open Loop Optimal Control of Buffer and Energy of a Wireless Device

Closed and Open Loop Optimal Control of Buffer and Energy of a Wireless Device Close an Open Loop Optimal Control of Buffer an Energy of a Wireless Device V. S. Borkar School of Technology an Computer Science TIFR, umbai, Inia. borkar@tifr.res.in A. A. Kherani B. J. Prabhu INRIA

More information

1. The electron volt is a measure of (A) charge (B) energy (C) impulse (D) momentum (E) velocity

1. The electron volt is a measure of (A) charge (B) energy (C) impulse (D) momentum (E) velocity AP Physics Multiple Choice Practice Electrostatics 1. The electron volt is a measure of (A) charge (B) energy (C) impulse (D) momentum (E) velocity. A soli conucting sphere is given a positive charge Q.

More information

Optimal Control of Spatially Distributed Systems

Optimal Control of Spatially Distributed Systems Optimal Control of Spatially Distribute Systems Naer Motee an Ali Jababaie Abstract In this paper, we stuy the structural properties of optimal control of spatially istribute systems. Such systems consist

More information

SYNCHRONOUS SEQUENTIAL CIRCUITS

SYNCHRONOUS SEQUENTIAL CIRCUITS CHAPTER SYNCHRONOUS SEUENTIAL CIRCUITS Registers an counters, two very common synchronous sequential circuits, are introuce in this chapter. Register is a igital circuit for storing information. Contents

More information

6.012 Electronic Devices and Circuits

6.012 Electronic Devices and Circuits Page 1 of 1 YOUR NAME Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology 6.12 Electronic Devices and Circuits Exam No. 1 Wednesday, October 7, 29 7:3 to 9:3

More information

IPMSM Inductances Calculation Using FEA

IPMSM Inductances Calculation Using FEA X International Symposium on Inustrial Electronics INDEL 24, Banja Luka, November 68, 24 IPMSM Inuctances Calculation Using FEA Dejan G. Jerkan, Marko A. Gecić an Darko P. Marčetić Department for Power,

More information

EXACT TRAVELING WAVE SOLUTIONS FOR A NEW NON-LINEAR HEAT TRANSFER EQUATION

EXACT TRAVELING WAVE SOLUTIONS FOR A NEW NON-LINEAR HEAT TRANSFER EQUATION THERMAL SCIENCE, Year 017, Vol. 1, No. 4, pp. 1833-1838 1833 EXACT TRAVELING WAVE SOLUTIONS FOR A NEW NON-LINEAR HEAT TRANSFER EQUATION by Feng GAO a,b, Xiao-Jun YANG a,b,* c, an Yu-Feng ZHANG a School

More information

IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 33, NO. 8, AUGUST

IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 33, NO. 8, AUGUST IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 33, NO. 8, AUGUST 214 1145 A Semi-Analytical Thermal Moeling Framework for Liqui-Coole ICs Arvin Srihar, Member, IEEE,

More information

Survey Sampling. 1 Design-based Inference. Kosuke Imai Department of Politics, Princeton University. February 19, 2013

Survey Sampling. 1 Design-based Inference. Kosuke Imai Department of Politics, Princeton University. February 19, 2013 Survey Sampling Kosuke Imai Department of Politics, Princeton University February 19, 2013 Survey sampling is one of the most commonly use ata collection methos for social scientists. We begin by escribing

More information

Efficient Macro-Micro Scale Coupled Modeling of Batteries

Efficient Macro-Micro Scale Coupled Modeling of Batteries A00 Journal of The Electrochemical Society, 15 10 A00-A008 005 0013-651/005/1510/A00/7/$7.00 The Electrochemical Society, Inc. Efficient Macro-Micro Scale Couple Moeling of Batteries Venkat. Subramanian,*,z

More information

We G Model Reduction Approaches for Solution of Wave Equations for Multiple Frequencies

We G Model Reduction Approaches for Solution of Wave Equations for Multiple Frequencies We G15 5 Moel Reuction Approaches for Solution of Wave Equations for Multiple Frequencies M.Y. Zaslavsky (Schlumberger-Doll Research Center), R.F. Remis* (Delft University) & V.L. Druskin (Schlumberger-Doll

More information

AN3400 Application note

AN3400 Application note Application note Analysis an simulation of a BJT complementary pair in a self-oscillating CFL solution ntrouction The steay-state oscillation of a novel zero-voltages switching (ZS) clampe-voltage (C)

More information

Compact Modeling of Graphene Barristor for Digital Integrated Circuit Design

Compact Modeling of Graphene Barristor for Digital Integrated Circuit Design Compact Moeling of Graphene Barristor for Digital Inteate Circuit Design Zhou Zhao, Xinlu Chen, Ashok Srivastava, Lu Peng Division of Electrical an Computer Engineering Louisiana State University Baton

More information

Nonlinear Dielectric Response of Periodic Composite Materials

Nonlinear Dielectric Response of Periodic Composite Materials onlinear Dielectric Response of Perioic Composite aterials A.G. KOLPAKOV 3, Bl.95, 9 th ovember str., ovosibirsk, 639 Russia the corresponing author e-mail: agk@neic.nsk.su, algk@ngs.ru A. K.TAGATSEV Ceramics

More information

Adjoint Transient Sensitivity Analysis in Circuit Simulation

Adjoint Transient Sensitivity Analysis in Circuit Simulation Ajoint Transient Sensitivity Analysis in Circuit Simulation Z. Ilievski 1, H. Xu 1, A. Verhoeven 1, E.J.W. ter Maten 1,2, W.H.A. Schilers 1,2 an R.M.M. Mattheij 1 1 Technische Universiteit Einhoven; e-mail:

More information

Generalization of the persistent random walk to dimensions greater than 1

Generalization of the persistent random walk to dimensions greater than 1 PHYSICAL REVIEW E VOLUME 58, NUMBER 6 DECEMBER 1998 Generalization of the persistent ranom walk to imensions greater than 1 Marián Boguñá, Josep M. Porrà, an Jaume Masoliver Departament e Física Fonamental,

More information

CONSERVATION PROPERTIES OF SMOOTHED PARTICLE HYDRODYNAMICS APPLIED TO THE SHALLOW WATER EQUATIONS

CONSERVATION PROPERTIES OF SMOOTHED PARTICLE HYDRODYNAMICS APPLIED TO THE SHALLOW WATER EQUATIONS BIT 0006-3835/00/4004-0001 $15.00 200?, Vol.??, No.??, pp.?????? c Swets & Zeitlinger CONSERVATION PROPERTIES OF SMOOTHE PARTICLE HYROYNAMICS APPLIE TO THE SHALLOW WATER EQUATIONS JASON FRANK 1 an SEBASTIAN

More information

Transmission Line Matrix (TLM) network analogues of reversible trapping processes Part B: scaling and consistency

Transmission Line Matrix (TLM) network analogues of reversible trapping processes Part B: scaling and consistency Transmission Line Matrix (TLM network analogues of reversible trapping processes Part B: scaling an consistency Donar e Cogan * ANC Eucation, 308-310.A. De Mel Mawatha, Colombo 3, Sri Lanka * onarecogan@gmail.com

More information

(3-3) = (Gauss s law) (3-6)

(3-3) = (Gauss s law) (3-6) tatic Electric Fiels Electrostatics is the stuy of the effects of electric charges at rest, an the static electric fiels, which are cause by stationary electric charges. In the euctive approach, few funamental

More information

Time-of-Arrival Estimation in Non-Line-Of-Sight Environments

Time-of-Arrival Estimation in Non-Line-Of-Sight Environments 2 Conference on Information Sciences an Systems, The Johns Hopkins University, March 2, 2 Time-of-Arrival Estimation in Non-Line-Of-Sight Environments Sinan Gezici, Hisashi Kobayashi an H. Vincent Poor

More information

An M/G/1 Retrial Queue with Priority, Balking and Feedback Customers

An M/G/1 Retrial Queue with Priority, Balking and Feedback Customers Journal of Convergence Information Technology Volume 5 Number April 1 An M/G/1 Retrial Queue with Priority Balking an Feeback Customers Peishu Chen * 1 Yiuan Zhu 1 * 1 Faculty of Science Jiangsu University

More information

Lecture 17 The Bipolar Junction Transistor (I) Forward Active Regime

Lecture 17 The Bipolar Junction Transistor (I) Forward Active Regime Lecture 17 The Bipolar Junction Transistor (I) Forward Active Regime Outline The Bipolar Junction Transistor (BJT): structure and basic operation I V characteristics in forward active regime Reading Assignment:

More information

Alpha Particle scattering

Alpha Particle scattering Introuction Alpha Particle scattering Revise Jan. 11, 014 In this lab you will stuy the interaction of α-particles ( 4 He) with matter, in particular energy loss an elastic scattering from a gol target

More information

Quantum Mechanics in Three Dimensions

Quantum Mechanics in Three Dimensions Physics 342 Lecture 20 Quantum Mechanics in Three Dimensions Lecture 20 Physics 342 Quantum Mechanics I Monay, March 24th, 2008 We begin our spherical solutions with the simplest possible case zero potential.

More information

figure shows a pnp transistor biased to operate in the active mode

figure shows a pnp transistor biased to operate in the active mode Lecture 10b EE-215 Electronic Devices and Circuits Asst Prof Muhammad Anis Chaudhary BJT: Device Structure and Physical Operation The pnp Transistor figure shows a pnp transistor biased to operate in the

More information

Electronic Devices and Circuit Theory

Electronic Devices and Circuit Theory Instructor s Resource Manual to accompany Electronic Devices an Circuit Theory Tenth Eition Robert L. Boylesta Louis Nashelsky Upper Sale River, New Jersey Columbus, Ohio Copyright 2009 by Pearson Eucation,

More information

Simple Electromagnetic Motor Model for Torsional Analysis of Variable Speed Drives with an Induction Motor

Simple Electromagnetic Motor Model for Torsional Analysis of Variable Speed Drives with an Induction Motor DOI: 10.24352/UB.OVGU-2017-110 TECHNISCHE MECHANIK, 37, 2-5, (2017), 347-357 submitte: June 15, 2017 Simple Electromagnetic Motor Moel for Torsional Analysis of Variable Spee Drives with an Inuction Motor

More information

ELEC3114 Control Systems 1

ELEC3114 Control Systems 1 ELEC34 Control Systems Linear Systems - Moelling - Some Issues Session 2, 2007 Introuction Linear systems may be represente in a number of ifferent ways. Figure shows the relationship between various representations.

More information

Thermal Modulation of Rayleigh-Benard Convection

Thermal Modulation of Rayleigh-Benard Convection Thermal Moulation of Rayleigh-Benar Convection B. S. Bhaauria Department of Mathematics an Statistics, Jai Narain Vyas University, Johpur, Inia-3400 Reprint requests to Dr. B. S.; E-mail: bsbhaauria@reiffmail.com

More information

ECE-342 Test 2 Solutions, Nov 4, :00-8:00pm, Closed Book (one page of notes allowed)

ECE-342 Test 2 Solutions, Nov 4, :00-8:00pm, Closed Book (one page of notes allowed) ECE-342 Test 2 Solutions, Nov 4, 2008 6:00-8:00pm, Closed Book (one page of notes allowed) Please use the following physical constants in your calculations: Boltzmann s Constant: Electron Charge: Free

More information

On some parabolic systems arising from a nuclear reactor model

On some parabolic systems arising from a nuclear reactor model On some parabolic systems arising from a nuclear reactor moel Kosuke Kita Grauate School of Avance Science an Engineering, Wasea University Introuction NR We stuy the following initial-bounary value problem

More information

Control of a PEM Fuel Cell Based on a Distributed Model

Control of a PEM Fuel Cell Based on a Distributed Model 21 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 3-July 2, 21 FrC1.6 Control of a PEM Fuel Cell Base on a Distribute Moel Michael Mangol Abstract To perform loa changes in proton

More information

arxiv: v1 [cond-mat.stat-mech] 9 Jan 2012

arxiv: v1 [cond-mat.stat-mech] 9 Jan 2012 arxiv:1201.1836v1 [con-mat.stat-mech] 9 Jan 2012 Externally riven one-imensional Ising moel Amir Aghamohammai a 1, Cina Aghamohammai b 2, & Mohamma Khorrami a 3 a Department of Physics, Alzahra University,

More information

Experimental Studies and Parametric Modeling of Ionic Flyers

Experimental Studies and Parametric Modeling of Ionic Flyers 1 Experimental Stuies an Parametric Moeling of Ionic Flyers Chor Fung Chung an Wen J. Li* Centre for Micro an Nano Systems, Faculty of Engineering The Chinese University of Hong Kong *Contact Author: wen@mae.cuhk.eu.hk

More information

SIMULATION OF POROUS MEDIUM COMBUSTION IN ENGINES

SIMULATION OF POROUS MEDIUM COMBUSTION IN ENGINES SIMULATION OF POROUS MEDIUM COMBUSTION IN ENGINES Jan Macek, Miloš Polášek Czech Technical University in Prague, Josef Božek Research Center Introuction Improvement of emissions from reciprocating internal

More information

Neural Network Training By Gradient Descent Algorithms: Application on the Solar Cell

Neural Network Training By Gradient Descent Algorithms: Application on the Solar Cell ISSN: 319-8753 Neural Networ Training By Graient Descent Algorithms: Application on the Solar Cell Fayrouz Dhichi*, Benyounes Ouarfi Department of Electrical Engineering, EEA&TI laboratory, Faculty of

More information

Convergence of Random Walks

Convergence of Random Walks Chapter 16 Convergence of Ranom Walks This lecture examines the convergence of ranom walks to the Wiener process. This is very important both physically an statistically, an illustrates the utility of

More information

The derivative of a function f(x) is another function, defined in terms of a limiting expression: f(x + δx) f(x)

The derivative of a function f(x) is another function, defined in terms of a limiting expression: f(x + δx) f(x) Y. D. Chong (2016) MH2801: Complex Methos for the Sciences 1. Derivatives The erivative of a function f(x) is another function, efine in terms of a limiting expression: f (x) f (x) lim x δx 0 f(x + δx)

More information

CLASS 3&4. BJT currents, parameters and circuit configurations

CLASS 3&4. BJT currents, parameters and circuit configurations CLASS 3&4 BJT currents, parameters and circuit configurations I E =I Ep +I En I C =I Cp +I Cn I B =I BB +I En -I Cn I BB =I Ep -I Cp I E = I B + I C I En = current produced by the electrons injected from

More information

TMA 4195 Matematisk modellering Exam Tuesday December 16, :00 13:00 Problems and solution with additional comments

TMA 4195 Matematisk modellering Exam Tuesday December 16, :00 13:00 Problems and solution with additional comments Problem F U L W D g m 3 2 s 2 0 0 0 0 2 kg 0 0 0 0 0 0 Table : Dimension matrix TMA 495 Matematisk moellering Exam Tuesay December 6, 2008 09:00 3:00 Problems an solution with aitional comments The necessary

More information

State of Charge Estimation of Cells in Series Connection by Using only the Total Voltage Measurement

State of Charge Estimation of Cells in Series Connection by Using only the Total Voltage Measurement 213 American Control Conference (ACC) Washington, DC, USA, June 17-19, 213 State of Charge Estimation of Cells in Series Connection by Using only the Total Voltage Measurement Xinfan Lin 1, Anna G. Stefanopoulou

More information

Hyperbolic Systems of Equations Posed on Erroneous Curved Domains

Hyperbolic Systems of Equations Posed on Erroneous Curved Domains Hyperbolic Systems of Equations Pose on Erroneous Curve Domains Jan Norström a, Samira Nikkar b a Department of Mathematics, Computational Mathematics, Linköping University, SE-58 83 Linköping, Sween (

More information

ensembles When working with density operators, we can use this connection to define a generalized Bloch vector: v x Tr x, v y Tr y

ensembles When working with density operators, we can use this connection to define a generalized Bloch vector: v x Tr x, v y Tr y Ph195a lecture notes, 1/3/01 Density operators for spin- 1 ensembles So far in our iscussion of spin- 1 systems, we have restricte our attention to the case of pure states an Hamiltonian evolution. Toay

More information

CURRENT ELECTRICITY Q.1

CURRENT ELECTRICITY Q.1 CUENT EECTCTY Q. Define Electric current an its unit.. Electric Current t can be efine as the time rate of flow of charge in a conuctor is calle Electric Current. The amount of flow of charge Q per unit

More information

Lectures - Week 10 Introduction to Ordinary Differential Equations (ODES) First Order Linear ODEs

Lectures - Week 10 Introduction to Ordinary Differential Equations (ODES) First Order Linear ODEs Lectures - Week 10 Introuction to Orinary Differential Equations (ODES) First Orer Linear ODEs When stuying ODEs we are consiering functions of one inepenent variable, e.g., f(x), where x is the inepenent

More information