Factorization of Indefinite Systems Associated with RLC Circuits. Patricio Rosen Wil Schilders, Joost Rommes
|
|
- Melinda Dixon
- 5 years ago
- Views:
Transcription
1 Factorization of Indefinite Systems Associated with RLC Circuits Patricio Rosen Wil Schilders, Joost Rommes
2 Outline Circuit Equations Solution Methods (Schilders Factorization Incidence Matrix Decomposition Schilders Type Factorizations for RL and RLC Performance as Direct Solver Conclusions /computer science and mathematics department PAGE
3 Motivation Small Size Complex Circuits Increasing Complexity Expensive Testing Circuit Simulation MOR is Necessary Solve the System /computer science and mathematics department PAGE 2
4 Circuit Equations System Formulation KVL: Av n =v b A = KCL: ATi b =0 A i A g A c A l,v b = v i v g v c v l,i b = i i i g ic il, i i =I t (t, i g = G v g, i c = Cd v, v c l = ( Ld +R i l, dt dt ( R Al ( ( il L ( 0 d il A T A TG + Ac v } l {{ g }}{{ n 0 A TC Ac }}{{ c dt v }}{{ n } G z(t C d z(t dt = ( 0 I AT t (t, }{{ i } B /computer science and mathematics department PAGE 3
5 Circuit Equations AC Analysis Complex System [( ˆR ˆP ˆPT Ĝ +iω ( ˆL 0 0 Ĉ ]( ir +ii i v r +iv i = ( 0 AT i Θ. Real System R ˆR ωˆl ωl PˆP 0 ωˆl ˆR 0 ˆP ˆPT 0 Ĝ ωĉ 0 ˆPT ωĉ Ĝ i r 0 ii 0 v = AT r i v i 0 Θ non symmetric and ( I 0 = PTA I ( A P PT D indefinite ( A 0 0 S ( I A P 0 I S= (D+PTA P /computer science and mathematics department PAGE 4
6 Circuit used for Eigenvalues /computer science and mathematics department PAGE 5
7 Circuit Equations Spectral Properties Complex Form Complex Stable Form /computer science and mathematics department PAGE 6
8 Circuit Equations Spectral Properties Real Form Real Stable Form /computer science and mathematics department PAGE 7
9 Solution Method
10 Solution Method Saddle Point Problem Indefinite Non symmetric ( ( A B x BT C y = ( b c Solution Methods: Direct Solvers Iterative Solvers expensive for large systems delay of convergence Paper by Greenbaum: Any nonincreasing convergence curve is possible for GMRES /computer science and mathematics department PAGE 9
11 Delay of Convergence Cx=r C= Rx=r R=r ij = N (0, /computer science and mathematics department PAGE 0
12 Schilders Factorization Invertible Symmetric Saddle Point A = ( Â ˆB ˆBT 0 ( x y = ( a b, Perform LQ Rearrange Matrix ΠˆB=BQ QAQ T =( A B BT 0 Q = ( Π 0 0 Q Schilders Factorization A = B 0 L B 2 I n m +L 2 M D 0 I m 0 D I m I m 0 0 A = LDL T B T BT I n m +LT 2 0 LT MT I m /computer science and mathematics department PAGE
13 Reference /computer science and mathematics department PAGE 2
14 RL Factorization
15 Incidence Matrix In general LQ decomposition ΠˆB=BQ We need only permutations Π ˆP =PΠ 2 Algorithm Idea: Π rˆpπc = ( x 0 v P Already Added Directly connected with Added Other Nodes Time Complexity: O(n2 /computer science and mathematics department PAGE 4
16 Remarks Lower Trapezoidal Form P P =( P 2 Inverse of top exists P = Calculation takes P in O(m2 Inverse is exact, it consistof 0,, /computer science and mathematics department PAGE 5
17 RL Factorization A = ˆR ωˆl ˆP 0 ωˆl ˆR 0 ˆP ˆPT ˆPT 0 0 QAQ T = P P = =( P 2 R ωl P 0 ωl R 0 P PT PT 0 0 Π 3 QAQ T ΠT 3 = R ωl R ωl2 2 P 0 ωl R ωl 2 R 2 0 P R ωl2 2 R ωl22 22 P 2 0 ωl 2 R 2 ωl 22 R 22 0 P 2 PT 0 PT PT 0 PT /computer science and mathematics department PAGE 6
18 RL Factorization Theorem: A A 2 B A 2 A 22 B 2 = B 0 L B 2 L 2 M D 0 I 2m 0 D 2 0 B T BT U 2 0 BT BT I 2 2m I 2m 0 0 U F I 2m Sketch of proof: B D BT +B U +L BT =A ( B D BT 2 +B F+L BT 2 =A 2 (2 B 2 D BT +B 2 U +MBT =A 2 (3 L 2 D 2 U 2 +B 2 D BT 2 +B 2 F+MBT 2 =A 22 (4 /computer science and mathematics department PAGE 7
19 Sketch of the proof From( D +U B T +B L = B A B T From(2 and (3 F =B ( A2 B D BT 2 L BT 2, M = ( A 2 B2 D BT B2 U B T. From (4 L 2 D 2 U 2 =A 22 B2 D BT 2 B2 F MBT 2 :=Ŵ If A is sym. pos.def. Computed with Cholesky RL Case: find LDU decomposition of Ŵ =(Ŵ Ŵ 2 Ŵ 2 Ŵ 22 /computer science and mathematics department PAGE 8
20 Sketch of the proof Lemma: Ŵ = ( L2, 0 ωl 2,2 L 2,3 (ω ( D2, ( L T ωl T 2, 2,2 D 2,2 (ω 0 LT (ω 2,3 Proof: PT 2 Ŵ =Ŵ ( ( ( R R P T W 2 =W 22 = P 2 P I 2 R 2 R 22 I Ŵ2 =Ŵ2 = ω ( P2 P I ( ( L L P T 2 L 2 L 22 I PT 2 Symmetric Positive Definite L 2, D 2, Cholesky L 2,2 =Ŵ2 L T 2, D 2, Solving L 2,3 D 2,2 LT 2,3 =Ŵ +ω 2Ŵ 2 Ŵ Ŵ 2 Cholesky /computer science and mathematics department PAGE 9
21 RL Factorization Final RL Factorization A= Π L DŨΠ T, L= I 0 0 2m L B 0, Ũ = M B 2 L 2 Frequency Dependencies: D,L,U,F,M,L 2,,D 2,,L 2,2 I U F 2m 0 BT BT U 2 ω D= 0 I 0 2m I 2m D 0, 0 0 D 2 independent or linearly dependent For different ω need to recompute only L 2,3 (ω,d 2,3 (ω /computer science and mathematics department PAGE 20
22 RLC FACTORIZATION
23 RLC Factorization Invertibility ( proof: A A B =, A= B T C ( ˆR Full rank ωˆl (ˆP 0, B= ωˆl ˆR 0 ˆP, C= (Ĝ ωĉ ωĉ Ĝ H= 2 (A+A T,D= 2 (C+C T Pos.Def 0=vTA v =xtax+xtbyt y TBTx+yTCy=xTHx+yTDy. G Rewrite Circuit Equations Ag i C g 0 R ic 0 Ac Al il + L 0 d dt A T g A T c A T l v n i g ic il v n = AT i I t (t /computer science and mathematics department PAGE 22
24 RLC Factorization ω>0 G 0 Ag 0 Ac R ωc ω L Al G 0 Ag C 0 Ac ω ω L R A l A T A T A T 0 0 g c l A T A T A T 0 0 g c l i gr i cr i lr i gi i ci i li v nr v ni A = ˆX Î(ωŶ ˆP 0 Î(ωŶ ˆX 0 ˆP ˆPT 0 0 0, 0 ˆPT 0 0 Î(ω= I I ω ωi /computer science and mathematics department PAGE 23
25 RLC Factorization Theorem: A A 2 B A 2 A 22 B 2 = B 0 L B 2 I 2(n m M D 0 I 2m 0 Ŵ 0 B T BT I 2(n m 0 BT BT I 2m I 2m 0 0 U F I 2m ŴW is invertible D 0 I 2m 0 Ŵ 0 I 2m 0 0 = 0 0 I 2m 0 Ŵ 0 I 2m 0 D Finish factorization with LDU Π e L 2 D 2 U 2 =Ŵ /computer science and mathematics department PAGE 24
26 RLC Factorization LDU decomposition Π e L 2 D 2 U 2 =Ŵ Theorem (Final RLC Factorization: A = Q T ΠT ΠT ΠT 3 E 4 L DŨΠ 4 Π Q 3 L= I 0 0 2m L B 0, Ũ = ΠTM ΠTB e e 2 L 2 I U F 2m 0 BT BT U 2 D= 0 I 0 2m I 2m D D 2 /computer science and mathematics department PAGE 25
27 Performance as Direct Solver
28 Performance as Direct Solver n ω : Frequencies m+ : Nodes n l : Resistor Inductor Branches n g : Conductances n c : Capacitors /computer science and mathematics department PAGE 27
29 Complexity of RL Algorithm RL Algorithm Rearrange Incidence Matrix P Find and D,L,U { O(m2 O(m3 best case worst case Perform Cholesky twice Ŵ W,Ŵ +ω2ŵ Ŵ,W W 2 W ŴW 2 2 O(n 3 l m 3 O Solve resulting systems ( 6 n ω 3 (n +m 3 l > O (( n ω O (( n ω + 5 (nl m 3 6 (nl m 3+3m3 best case worst case RL IS ALWAYS BETTER THAN LU DECOMPOSITION /computer science and mathematics department PAGE 28
30 RL example L = I N pi N pi N I N M3 M3= p p p p N: number of blocks p: coupling factor /computer science and mathematics department PAGE 29
31 Running times No Coupling Coupling of 0% /computer science and mathematics department PAGE 30
32 RL performs better It does not perform that nice due to: P P P P T We are in the worst case complexity /computer science and mathematics department PAGE 3
33 Running times modified circuit Modified Circuit No Coupling Coupling of 0% /computer science and mathematics department PAGE 32
34 RLC Algorithm
35 Complexity of RLC Algorithm RLC Algorithm Find G, C Perform LU decomposition Ŵ RLC Fact O((n w +4m3+n w 6 3 (n l +n g +n c O ( n w 6 3 (n l +n g +n c m 3 m 3 worst case best case LU O ( n 6 (n +m 3 ω 3 l if n g +n c 2m Conditionally Better than LU /computer science and mathematics department PAGE 34
36 RLC Ladder Example N: number of blocks /computer science and mathematics department PAGE 35
37 Running times LU is better than RLC factorization /computer science and mathematics department PAGE 36
38 Modified RLC Ladder RLC factorization is still usefull Modified RLC Ladder circuit: 0 p pn Remove: capacitor branches from left to right pn and conductances from right to left /computer science and mathematics department PAGE 37
39 P P P P T /computer science and mathematics department PAGE 38
40 Running Times Modified RLC Ladder 50% Conductances and Capacitors /computer science and mathematics department PAGE 39
41 Running Times Modified Circuit 20% Conductances and Capacitors /computer science and mathematics department PAGE 40
42 Running Times Modified Circuit RLC is better than LU if: n g +n c 2m /computer science and mathematics department PAGE 4
43 Conclusions Results: Explicit Factorizations RL/RLC system Frequency dependencies founded RL algorithm always better than LU decomposition RLC algorihtm conditionally better than LU LU needs to recompute all again RLC only need to recompute some parts Future work: Control Fill-in Study non-linear frequency dependencies /computer science and mathematics department PAGE 42
44 THANKS FOR YOUR ATTENTION
45 L = Frequency Dependencies RL Case D = U = F = ( diag(p R P T diag(p R P T ( P strlow(p R P T 0 ωl P T P strlow(p R P T ( strupp(p R P T PT ωp L 0 strupp(p R P TPT ( P R 2 low(p R P T PT ωp 2 L 2 P L P TPT P R 2 2 low(p ω(p L 2 R P T PT 2 M = ( R2 P T P2 upp(p ωl 2 P T R P T ω(l2 P T R 2 P T +P 2 P L P T R P T P2 upp(p /computer science and mathematics department PAGE 44
46 F = M = Frequency Dependencies RLC Case ( diag(p X D = P T 0 0 diag(p X P, T ( P strlow(p X L = P T 0 I (ωy P T P strlow(p X P, T ( strupp(p X U = P TPT P Y I (ω 0 strupp(p X P. TPT ( P X 2 low(p X P TPT P Y 2 2 I 2 (ω P Y 2 I 2 (ω+p I (ωy P TPT P X 2 2 low(p ( X2 P T P2 upp(p X P T I 2 (ωy 2 P T P2 P I2 (ωy 2 P T X 2 P T P2 upp(p X P, TPT 2 Y I (ωp T X P. T /computer science and mathematics department PAGE 45
47 Circuit Equations Incidence Matrix Kirchhoff s Current Law Kirchhoff s Voltage Law Branch Constitutive Relations A= A i A g A c A l Av i n =v, i, A g v n =v g, A c v n =v c, Av l n =v l A, i b, v b, v n v i v g v c v l,v b =,i b = i i i g ic il AT i i i +AT g i g +AT c i c +AT l i l =0 ATi b =0 Av n =v b G, C, R diagonal pos. def. L symmetricpos.def. i i =I t (t, i g = G v g, i c = Cd dt v c, v l = ( Ld dt +R i l, /computer science and mathematics department PAGE 46
48 Circuit Equations Alternate Current Analysis ( R Al A T A TG Ac } l {{ g } G ( il v n Consider Wave Input ( L A TC Ac }{{ c } C d dt ( il v n G(Z(weiωt+C d dt (Z(ωe iωt=bθeiωt, = ( 0 I AT t (t }{{ i } B GZ(w+iωCZ(w=BΘ [( ˆR ˆP ˆPT Ĝ +iω ( ˆL 0 0 Ĉ ]( ir +ii i v r +iv i = ( 0 AT i Θ. /computer science and mathematics department PAGE 47
49 Schilders Factorization Permuting Useful for: Direct Solver Preconditioner A = QLDL T Q T, ( Q 0 Π T = QT 0 L = I 0 0 m L B 0 M B I +L 2 n m 2 D = 0 I 0 m I m D D 2 Goal: develop Schilders type factorizations for the Circuit Equations /computer science and mathematics department PAGE 48
50 Simulation for N=2 Magnitude Phase /computer science and mathematics department PAGE 49
51 Circuit Equations A circuit is a network of interconnected components Circuit topology Incidence Matrix Set ground node g A= N N 2 N {}} 3 { B B 2 B 3 /computer science and mathematics department PAGE 50
52 Introduction Electronics Industry Circuit Design Circuit Simulation /computer science and mathematics department PAGE 5
Eindhoven University of Technology MASTER. Factorization of indefinite systems associated with RLC circuits. Rosen Esquivel, P.I.
Eindhoven University of Technology MASTER Factorization of indefinite systems associated with RLC circuits Rosen Esquivel, P.I. Award date: 28 Link to publication Disclaimer This document contains a student
More informationModel order reduction of electrical circuits with nonlinear elements
Model order reduction of electrical circuits with nonlinear elements Andreas Steinbrecher and Tatjana Stykel 1 Introduction The efficient and robust numerical simulation of electrical circuits plays a
More informationNetwork Graphs and Tellegen s Theorem
Networ Graphs and Tellegen s Theorem The concepts of a graph Cut sets and Kirchhoff s current laws Loops and Kirchhoff s voltage laws Tellegen s Theorem The concepts of a graph The analysis of a complex
More informationElectric Circuits I. Nodal Analysis. Dr. Firas Obeidat
Electric Circuits I Nodal Analysis Dr. Firas Obeidat 1 Nodal Analysis Without Voltage Source Nodal analysis, which is based on a systematic application of Kirchhoff s current law (KCL). A node is defined
More informationIdentification of Electrical Circuits for Realization of Sparsity Preserving Reduced Order Models
Identification of Electrical Circuits for Realization of Sparsity Preserving Reduced Order Models Christof Kaufmann 25th March 2010 Abstract Nowadays very-large scale integrated circuits contain a large
More informationBasic RL and RC Circuits R-L TRANSIENTS: STORAGE CYCLE. Engineering Collage Electrical Engineering Dep. Dr. Ibrahim Aljubouri
st Class Basic RL and RC Circuits The RL circuit with D.C (steady state) The inductor is short time at Calculate the inductor current for circuits shown below. I L E R A I L E R R 3 R R 3 I L I L R 3 R
More informationAC Circuit Analysis and Measurement Lab Assignment 8
Electric Circuit Lab Assignments elcirc_lab87.fm - 1 AC Circuit Analysis and Measurement Lab Assignment 8 Introduction When analyzing an electric circuit that contains reactive components, inductors and
More informationOutline. Week 5: Circuits. Course Notes: 3.5. Goals: Use linear algebra to determine voltage drops and branch currents.
Outline Week 5: Circuits Course Notes: 3.5 Goals: Use linear algebra to determine voltage drops and branch currents. Components in Resistor Networks voltage source current source resistor Components in
More informationKirchhoff's Laws and Circuit Analysis (EC 2)
Kirchhoff's Laws and Circuit Analysis (EC ) Circuit analysis: solving for I and V at each element Linear circuits: involve resistors, capacitors, inductors Initial analysis uses only resistors Power sources,
More informationLecture #3. Review: Power
Lecture #3 OUTLINE Power calculations Circuit elements Voltage and current sources Electrical resistance (Ohm s law) Kirchhoff s laws Reading Chapter 2 Lecture 3, Slide 1 Review: Power If an element is
More informationNetwork Topology-2 & Dual and Duality Choice of independent branch currents and voltages: The solution of a network involves solving of all branch currents and voltages. We know that the branch current
More informationA geometric Birkhoffian formalism for nonlinear RLC networks
Journal of Geometry and Physics 56 (2006) 2545 2572 www.elsevier.com/locate/jgp A geometric Birkhoffian formalism for nonlinear RLC networks Delia Ionescu Institute of Mathematics, Romanian Academy of
More informationSolution of indefinite linear systems using an LQ decomposition for the linear constraints Schilders, W.H.A.
Solution of indefinite linear systems using an LQ decomposition for the linear constraints Schilders, W.H.A. Published: 01/01/2009 Document Version Publisher s PDF, also known as Version of Record includes
More informationSinusoidal Steady State Analysis (AC Analysis) Part I
Sinusoidal Steady State Analysis (AC Analysis) Part I Amin Electronics and Electrical Communications Engineering Department (EECE) Cairo University elc.n102.eng@gmail.com http://scholar.cu.edu.eg/refky/
More informationBasics of Network Theory (Part-I)
Basics of Network Theory (PartI). A square waveform as shown in figure is applied across mh ideal inductor. The current through the inductor is a. wave of peak amplitude. V 0 0.5 t (m sec) [Gate 987: Marks]
More informationChapter 33. Alternating Current Circuits
Chapter 33 Alternating Current Circuits 1 Capacitor Resistor + Q = C V = I R R I + + Inductance d I Vab = L dt AC power source The AC power source provides an alternative voltage, Notation - Lower case
More informationJacobi-Davidson methods and preconditioning with applications in pole-zero analysis
Nat.Lab. Unclassified Report 2002/817 Date of issue: 05/2002 Jacobi-Davidson methods and preconditioning with applications in pole-zero analysis Master s Thesis Joost Rommes Unclassified Report 2002/817
More informationChapter 2. Engr228 Circuit Analysis. Dr Curtis Nelson
Chapter 2 Engr228 Circuit Analysis Dr Curtis Nelson Chapter 2 Objectives Understand symbols and behavior of the following circuit elements: Independent voltage and current sources; Dependent voltage and
More informationTo find the step response of an RC circuit
To find the step response of an RC circuit v( t) v( ) [ v( t) v( )] e tt The time constant = RC The final capacitor voltage v() The initial capacitor voltage v(t ) To find the step response of an RL circuit
More informationLAPLACE TRANSFORMATION AND APPLICATIONS. Laplace transformation It s a transformation method used for solving differential equation.
LAPLACE TRANSFORMATION AND APPLICATIONS Laplace transformation It s a transformation method used for solving differential equation. Advantages The solution of differential equation using LT, progresses
More informationLINEAR ALGEBRA QUESTION BANK
LINEAR ALGEBRA QUESTION BANK () ( points total) Circle True or False: TRUE / FALSE: If A is any n n matrix, and I n is the n n identity matrix, then I n A = AI n = A. TRUE / FALSE: If A, B are n n matrices,
More informationElectrical Circuits (2)
Electrical Circuits (2) Lecture 7 Transient Analysis Dr.Eng. Basem ElHalawany Extra Reference for this Lecture Chapter 16 Schaum's Outline Of Theory And Problems Of Electric Circuits https://archive.org/details/theoryandproblemsofelectriccircuits
More informationStability and Passivity of the Super Node Algorithm for EM Modeling of IC s
Stability and Passivity of the Super Node Algorithm for EM Modeling of IC s M.V. Ugryumova and W.H.A. Schilders Abstract The super node algorithm performs model order reduction based on physical principles.
More informationModel reduction of nonlinear circuit equations
Model reduction of nonlinear circuit equations Tatjana Stykel Technische Universität Berlin Joint work with T. Reis and A. Steinbrecher BIRS Workshop, Banff, Canada, October 25-29, 2010 T. Stykel. Model
More informationSolution: Based on the slope of q(t): 20 A for 0 t 1 s dt = 0 for 3 t 4 s. 20 A for 4 t 5 s 0 for t 5 s 20 C. t (s) 20 C. i (A) Fig. P1.
Problem 1.24 The plot in Fig. P1.24 displays the cumulative charge q(t) that has entered a certain device up to time t. Sketch a plot of the corresponding current i(t). q 20 C 0 1 2 3 4 5 t (s) 20 C Figure
More informationHandout 10: Inductance. Self-Inductance and inductors
1 Handout 10: Inductance Self-Inductance and inductors In Fig. 1, electric current is present in an isolate circuit, setting up magnetic field that causes a magnetic flux through the circuit itself. This
More informationAutomatic Formulation of Circuit Equations
ECE 570 Session 3 IC 752-E Computer Aided Engineering for Integrated Circuits Automatic Formulation of Circuit Equations Objective: Basics of computer aided analysis/simulation Outline:. Discussion of
More informationSeries & Parallel Resistors 3/17/2015 1
Series & Parallel Resistors 3/17/2015 1 Series Resistors & Voltage Division Consider the single-loop circuit as shown in figure. The two resistors are in series, since the same current i flows in both
More informationHandout 11: AC circuit. AC generator
Handout : AC circuit AC generator Figure compares the voltage across the directcurrent (DC) generator and that across the alternatingcurrent (AC) generator For DC generator, the voltage is constant For
More informationSINUSOIDAL STEADY STATE CIRCUIT ANALYSIS
SINUSOIDAL STEADY STATE CIRCUIT ANALYSIS 1. Introduction A sinusoidal current has the following form: where I m is the amplitude value; ω=2 πf is the angular frequency; φ is the phase shift. i (t )=I m.sin
More informationSinusoidal Response of RLC Circuits
Sinusoidal Response of RLC Circuits Series RL circuit Series RC circuit Series RLC circuit Parallel RL circuit Parallel RC circuit R-L Series Circuit R-L Series Circuit R-L Series Circuit Instantaneous
More informationPhysics 115. AC: RL vs RC circuits Phase relationships RLC circuits. General Physics II. Session 33
Session 33 Physics 115 General Physics II AC: RL vs RC circuits Phase relationships RLC circuits R. J. Wilkes Email: phy115a@u.washington.edu Home page: http://courses.washington.edu/phy115a/ 6/2/14 1
More informationQUESTION BANK SUBJECT: NETWORK ANALYSIS (10ES34)
QUESTION BANK SUBJECT: NETWORK ANALYSIS (10ES34) NOTE: FOR NUMERICAL PROBLEMS FOR ALL UNITS EXCEPT UNIT 5 REFER THE E-BOOK ENGINEERING CIRCUIT ANALYSIS, 7 th EDITION HAYT AND KIMMERLY. PAGE NUMBERS OF
More informationEE 40: Introduction to Microelectronic Circuits Spring 2008: Midterm 2
EE 4: Introduction to Microelectronic Circuits Spring 8: Midterm Venkat Anantharam 3/9/8 Total Time Allotted : min Total Points:. This is a closed book exam. However, you are allowed to bring two pages
More informationFigure Circuit for Question 1. Figure Circuit for Question 2
Exercises 10.7 Exercises Multiple Choice 1. For the circuit of Figure 10.44 the time constant is A. 0.5 ms 71.43 µs 2, 000 s D. 0.2 ms 4 Ω 2 Ω 12 Ω 1 mh 12u 0 () t V Figure 10.44. Circuit for Question
More informationMAT292 - Calculus III - Fall Solution for Term Test 2 - November 6, 2014 DO NOT WRITE ON THE QR CODE AT THE TOP OF THE PAGES.
MAT9 - Calculus III - Fall 4 Solution for Term Test - November 6, 4 Time allotted: 9 minutes. Aids permitted: None. Full Name: Last First Student ID: Email: @mail.utoronto.ca Instructions DO NOT WRITE
More informationModel Order Reduction for Electronic Circuits: Mathematical and Physical Approaches
Proceedings of the 2nd Fields MITACS Industrial Problem-Solving Workshop, 2008 Model Order Reduction for Electronic Circuits: Mathematical and Physical Approaches Problem Presenter: Wil Schilders, NXP
More informationENGR 2405 Class No Electric Circuits I
ENGR 2405 Class No. 48056 Electric Circuits I Dr. R. Williams Ph.D. rube.williams@hccs.edu Electric Circuit An electric circuit is an interconnec9on of electrical elements Charge Charge is an electrical
More informationarxiv:math/ v1 [math.ds] 5 Sep 2006
arxiv:math/0609153v1 math.ds 5 Sep 2006 A geometric Birkhoffian formalism for nonlinear RLC networks Delia Ionescu, Institute of Mathematics of the Romanian Academy P.O. Box 1-764, RO-014700, Bucharest,
More informationSinusoidal Steady-State Analysis
Chapter 4 Sinusoidal Steady-State Analysis In this unit, we consider circuits in which the sources are sinusoidal in nature. The review section of this unit covers most of section 9.1 9.9 of the text.
More informationP441 Analytical Mechanics - I. RLC Circuits. c Alex R. Dzierba. In this note we discuss electrical oscillating circuits: undamped, damped and driven.
Lecture 10 Monday - September 19, 005 Written or last updated: September 19, 005 P441 Analytical Mechanics - I RLC Circuits c Alex R. Dzierba Introduction In this note we discuss electrical oscillating
More informationSymmetric matrices and dot products
Symmetric matrices and dot products Proposition An n n matrix A is symmetric iff, for all x, y in R n, (Ax) y = x (Ay). Proof. If A is symmetric, then (Ax) y = x T A T y = x T Ay = x (Ay). If equality
More informationREACTANCE. By: Enzo Paterno Date: 03/2013
REACTANCE REACTANCE By: Enzo Paterno Date: 03/2013 5/2007 Enzo Paterno 1 RESISTANCE - R i R (t R A resistor for all practical purposes is unaffected by the frequency of the applied sinusoidal voltage or
More informationAlgebra C Numerical Linear Algebra Sample Exam Problems
Algebra C Numerical Linear Algebra Sample Exam Problems Notation. Denote by V a finite-dimensional Hilbert space with inner product (, ) and corresponding norm. The abbreviation SPD is used for symmetric
More informationFirst-order transient
EIE209 Basic Electronics First-order transient Contents Inductor and capacitor Simple RC and RL circuits Transient solutions Constitutive relation An electrical element is defined by its relationship between
More informationTHE INVERSE BOUNDARY PROBLEM FOR GENERAL PLANAR ELECTRICAL NETWORKS
THE INVERSE BOUNDARY PROBLEM FOR GENERAL PLANAR ELECTRICAL NETWORKS DMITRIY LEYKEKHMAN Abstract. We consider an electrical network where each edge is consists of resistor, inductor, and capacitor joined
More informationThis can be accomplished by left matrix multiplication as follows: I
1 Numerical Linear Algebra 11 The LU Factorization Recall from linear algebra that Gaussian elimination is a method for solving linear systems of the form Ax = b, where A R m n and bran(a) In this method
More informationPositive Definite Matrix
1/29 Chia-Ping Chen Professor Department of Computer Science and Engineering National Sun Yat-sen University Linear Algebra Positive Definite, Negative Definite, Indefinite 2/29 Pure Quadratic Function
More informationA Brief Outline of Math 355
A Brief Outline of Math 355 Lecture 1 The geometry of linear equations; elimination with matrices A system of m linear equations with n unknowns can be thought of geometrically as m hyperplanes intersecting
More informationforms Christopher Engström November 14, 2014 MAA704: Matrix factorization and canonical forms Matrix properties Matrix factorization Canonical forms
Christopher Engström November 14, 2014 Hermitian LU QR echelon Contents of todays lecture Some interesting / useful / important of matrices Hermitian LU QR echelon Rewriting a as a product of several matrices.
More informationĞ ğ ğ Ğ ğ Öğ ç ğ ö öğ ğ ŞÇ ğ ğ
Ğ Ü Ü Ü ğ ğ ğ Öğ ş öğ ş ğ öğ ö ö ş ğ ğ ö ğ Ğ ğ ğ Ğ ğ Öğ ç ğ ö öğ ğ ŞÇ ğ ğ l _.j l L., c :, c Ll Ll, c :r. l., }, l : ö,, Lc L.. c l Ll Lr. 0 c (} >,! l LA l l r r l rl c c.r; (Y ; c cy c r! r! \. L : Ll.,
More informationSource-Free RC Circuit
First Order Circuits Source-Free RC Circuit Initial charge on capacitor q = Cv(0) so that voltage at time 0 is v(0). What is v(t)? Prof Carruthers (ECE @ BU) EK307 Notes Summer 2018 150 / 264 First Order
More informationEIT Review. Electrical Circuits DC Circuits. Lecturer: Russ Tatro. Presented by Tau Beta Pi The Engineering Honor Society 10/3/2006 1
EIT Review Electrical Circuits DC Circuits Lecturer: Russ Tatro Presented by Tau Beta Pi The Engineering Honor Society 10/3/2006 1 Session Outline Basic Concepts Basic Laws Methods of Analysis Circuit
More informationResponse of Second-Order Systems
Unit 3 Response of SecondOrder Systems In this unit, we consider the natural and step responses of simple series and parallel circuits containing inductors, capacitors and resistors. The equations which
More informationSinusoidal Steady-State Analysis
Sinusoidal Steady-State Analysis Mauro Forti October 27, 2018 Constitutive Relations in the Frequency Domain Consider a network with independent voltage and current sources at the same angular frequency
More informationLECTURE 8 RC AND RL FIRST-ORDER CIRCUITS (PART 1)
CIRCUITS by Ulaby & Maharbiz LECTURE 8 RC AND RL FIRST-ORDER CIRCUITS (PART 1) 07/18/2013 ECE225 CIRCUIT ANALYSIS All rights reserved. Do not copy or distribute. 2013 National Technology and Science Press
More informationA Chebyshev-based two-stage iterative method as an alternative to the direct solution of linear systems
A Chebyshev-based two-stage iterative method as an alternative to the direct solution of linear systems Mario Arioli m.arioli@rl.ac.uk CCLRC-Rutherford Appleton Laboratory with Daniel Ruiz (E.N.S.E.E.I.H.T)
More informationA~(A'~) = i,(t) (1.34)
GENERAL RESISTIVE CIRCUITS 225 Nonlinear branch equation In vector notation, Eq. (1.31) becomes simply Since the independent current sources do not form cut sets (by assumption), Eq. (1.14) remains valid.
More informationImportant Matrix Factorizations
LU Factorization Choleski Factorization The QR Factorization LU Factorization: Gaussian Elimination Matrices Gaussian elimination transforms vectors of the form a α, b where a R k, 0 α R, and b R n k 1,
More information2.1 The electric field outside a charged sphere is the same as for a point source, E(r) =
Chapter Exercises. The electric field outside a charged sphere is the same as for a point source, E(r) Q 4πɛ 0 r, where Q is the charge on the inner surface of radius a. The potential drop is the integral
More information18.06 Professor Johnson Quiz 1 October 3, 2007
18.6 Professor Johnson Quiz 1 October 3, 7 SOLUTIONS 1 3 pts.) A given circuit network directed graph) which has an m n incidence matrix A rows = edges, columns = nodes) and a conductance matrix C [diagonal
More information15 n=0. zz = re jθ re jθ = r 2. (b) For division and multiplication, it is handy to use the polar representation: z = rejθ. = z 1z 2.
Professor Fearing EECS0/Problem Set v.0 Fall 06 Due at 4 pm, Fri. Sep. in HW box under stairs (st floor Cory) Reading: EE6AB notes. This problem set should be review of material from EE6AB. (Please note,
More information1 Phasors and Alternating Currents
Physics 4 Chapter : Alternating Current 0/5 Phasors and Alternating Currents alternating current: current that varies sinusoidally with time ac source: any device that supplies a sinusoidally varying potential
More information9. Introduction and Chapter Objectives
Real Analog - Circuits 1 Chapter 9: Introduction to State Variable Models 9. Introduction and Chapter Objectives In our analysis approach of dynamic systems so far, we have defined variables which describe
More informationCURRENT SOURCES EXAMPLE 1 Find the source voltage Vs and the current I1 for the circuit shown below SOURCE CONVERSIONS
CURRENT SOURCES EXAMPLE 1 Find the source voltage Vs and the current I1 for the circuit shown below EXAMPLE 2 Find the source voltage Vs and the current I1 for the circuit shown below SOURCE CONVERSIONS
More informationThe antitriangular factorisation of saddle point matrices
The antitriangular factorisation of saddle point matrices J. Pestana and A. J. Wathen August 29, 2013 Abstract Mastronardi and Van Dooren [this journal, 34 (2013) pp. 173 196] recently introduced the block
More informationEE221 Circuits II. Chapter 14 Frequency Response
EE22 Circuits II Chapter 4 Frequency Response Frequency Response Chapter 4 4. Introduction 4.2 Transfer Function 4.3 Bode Plots 4.4 Series Resonance 4.5 Parallel Resonance 4.6 Passive Filters 4.7 Active
More informationIntroduction to Scientific Computing
(Lecture 5: Linear system of equations / Matrix Splitting) Bojana Rosić, Thilo Moshagen Institute of Scientific Computing Motivation Let us resolve the problem scheme by using Kirchhoff s laws: the algebraic
More informationEE292: Fundamentals of ECE
EE292: Fundamentals of ECE Fall 2012 TTh 10:00-11:15 SEB 1242 Lecture 20 121101 http://www.ee.unlv.edu/~b1morris/ee292/ 2 Outline Chapters 1-3 Circuit Analysis Techniques Chapter 10 Diodes Ideal Model
More informationEXAMPLES OF CLASSICAL ITERATIVE METHODS
EXAMPLES OF CLASSICAL ITERATIVE METHODS In these lecture notes we revisit a few classical fixpoint iterations for the solution of the linear systems of equations. We focus on the algebraic and algorithmic
More informationLecture 6. Eigen-analysis
Lecture 6 Eigen-analysis University of British Columbia, Vancouver Yue-Xian Li March 7 6 Definition of eigenvectors and eigenvalues Def: Any n n matrix A defines a LT, A : R n R n A vector v and a scalar
More informationEE221 Circuits II. Chapter 14 Frequency Response
EE22 Circuits II Chapter 4 Frequency Response Frequency Response Chapter 4 4. Introduction 4.2 Transfer Function 4.3 Bode Plots 4.4 Series Resonance 4.5 Parallel Resonance 4.6 Passive Filters 4.7 Active
More informationLecture 11 - AC Power
- AC Power 11/17/2015 Reading: Chapter 11 1 Outline Instantaneous power Complex power Average (real) power Reactive power Apparent power Maximum power transfer Power factor correction 2 Power in AC Circuits
More informationStructured Preconditioners for Saddle Point Problems
Structured Preconditioners for Saddle Point Problems V. Simoncini Dipartimento di Matematica Università di Bologna valeria@dm.unibo.it p. 1 Collaborators on this project Mario Arioli, RAL, UK Michele Benzi,
More informationPreamble. Circuit Analysis II. Mesh Analysis. When circuits get really complex methods learned so far will still work,
Preamble Circuit Analysis II Physics, 8 th Edition Custom Edition Cutnell & Johnson When circuits get really complex methods learned so far will still work, but they can take a long time to do. A particularly
More informationI(t) R L. RL Circuit: Fundamentals. a b. Specifications: E (emf) R (resistance) L (inductance) Switch S: a: current buildup. b: current shutdown
RL Circuit: Fundamentals pecifications: E (emf) R (resistance) L (inductance) witch : a: current buildup a b I(t) R L b: current shutdown Time-dependent quantities: I(t): instantaneous current through
More informationVariational Integrators for Electrical Circuits
Variational Integrators for Electrical Circuits Sina Ober-Blöbaum California Institute of Technology Joint work with Jerrold E. Marsden, Houman Owhadi, Molei Tao, and Mulin Cheng Structured Integrators
More informationRLC Series Circuit. We can define effective resistances for capacitors and inductors: 1 = Capacitive reactance:
RLC Series Circuit In this exercise you will investigate the effects of changing inductance, capacitance, resistance, and frequency on an RLC series AC circuit. We can define effective resistances for
More informationElectric Circuits Fall 2015 Solution #5
RULES: Please try to work on your own. Discussion is permissible, but identical submissions are unacceptable! Please show all intermeate steps: a correct solution without an explanation will get zero cret.
More informationECE 1311: Electric Circuits. Chapter 2: Basic laws
ECE 1311: Electric Circuits Chapter 2: Basic laws Basic Law Overview Ideal sources series and parallel Ohm s law Definitions open circuits, short circuits, conductance, nodes, branches, loops Kirchhoff's
More informationAnnouncements: Today: more AC circuits
Announcements: Today: more AC circuits I 0 I rms Current through a light bulb I 0 I rms I t = I 0 cos ωt I 0 Current through a LED I t = I 0 cos ωt Θ(cos ωt ) Theta function (is zero for a negative argument)
More informationMay 9, 2014 MATH 408 MIDTERM EXAM OUTLINE. Sample Questions
May 9, 24 MATH 48 MIDTERM EXAM OUTLINE This exam will consist of two parts and each part will have multipart questions. Each of the 6 questions is worth 5 points for a total of points. The two part of
More informationAn Efficient Graph Sparsification Approach to Scalable Harmonic Balance (HB) Analysis of Strongly Nonlinear RF Circuits
Design Automation Group An Efficient Graph Sparsification Approach to Scalable Harmonic Balance (HB) Analysis of Strongly Nonlinear RF Circuits Authors : Lengfei Han (Speaker) Xueqian Zhao Dr. Zhuo Feng
More informationIndefinite Preconditioners for PDE-constrained optimization problems. V. Simoncini
Indefinite Preconditioners for PDE-constrained optimization problems V. Simoncini Dipartimento di Matematica, Università di Bologna, Italy valeria.simoncini@unibo.it Partly joint work with Debora Sesana,
More informationComputational Economics and Finance
Computational Economics and Finance Part II: Linear Equations Spring 2016 Outline Back Substitution, LU and other decomposi- Direct methods: tions Error analysis and condition numbers Iterative methods:
More informationNote 11: Alternating Current (AC) Circuits
Note 11: Alternating Current (AC) Circuits V R No phase difference between the voltage difference and the current and max For alternating voltage Vmax sin t, the resistor current is ir sin t. the instantaneous
More informationSome Reviews on Ranks of Upper Triangular Block Matrices over a Skew Field
International Mathematical Forum, Vol 13, 2018, no 7, 323-335 HIKARI Ltd, wwwm-hikaricom https://doiorg/1012988/imf20188528 Some Reviews on Ranks of Upper Triangular lock Matrices over a Skew Field Netsai
More informationHOMEWORK 4: MATH 265: SOLUTIONS. y p = cos(ω 0t) 9 ω 2 0
HOMEWORK 4: MATH 265: SOLUTIONS. Find the solution to the initial value problems y + 9y = cos(ωt) with y(0) = 0, y (0) = 0 (account for all ω > 0). Draw a plot of the solution when ω = and when ω = 3.
More informationv = 1(1 t) + 1(1 + t). We may consider these components as vectors in R n and R m : w 1. R n, w m
20 Diagonalization Let V and W be vector spaces, with bases S = {e 1,, e n } and T = {f 1,, f m } respectively Since these are bases, there exist constants v i and w such that any vectors v V and w W can
More informationOscillations and Electromagnetic Waves. March 30, 2014 Chapter 31 1
Oscillations and Electromagnetic Waves March 30, 2014 Chapter 31 1 Three Polarizers! Consider the case of unpolarized light with intensity I 0 incident on three polarizers! The first polarizer has a polarizing
More informationBlockMatrixComputations and the Singular Value Decomposition. ATaleofTwoIdeas
BlockMatrixComputations and the Singular Value Decomposition ATaleofTwoIdeas Charles F. Van Loan Department of Computer Science Cornell University Supported in part by the NSF contract CCR-9901988. Block
More informationThe RLC circuits have a wide range of applications, including oscillators and frequency filters
9. The RL ircuit The RL circuits have a wide range of applications, including oscillators and frequency filters This chapter considers the responses of RL circuits The result is a second-order differential
More informationEECS2200 Electric Circuits. RLC Circuit Natural and Step Responses
5--4 EECS Electric Circuit Chapter 6 R Circuit Natural and Step Repone Objective Determine the repone form of the circuit Natural repone parallel R circuit Natural repone erie R circuit Step repone of
More informationElectric Circuit Theory
Electric Circuit Theory Nam Ki Min nkmin@korea.ac.kr 010-9419-2320 Chapter 11 Sinusoidal Steady-State Analysis Nam Ki Min nkmin@korea.ac.kr 010-9419-2320 Contents and Objectives 3 Chapter Contents 11.1
More informationET3-7: Modelling II(V) Electrical, Mechanical and Thermal Systems
ET3-7: Modelling II(V) Electrical, Mechanical and Thermal Systems Agenda of the Day 1. Resume of lesson I 2. Basic system models. 3. Models of basic electrical system elements 4. Application of Matlab/Simulink
More information4/27 Friday. I have all the old homework if you need to collect them.
4/27 Friday Last HW: do not need to turn it. Solution will be posted on the web. I have all the old homework if you need to collect them. Final exam: 7-9pm, Monday, 4/30 at Lambert Fieldhouse F101 Calculator
More informationInductance, Inductors, RL Circuits & RC Circuits, LC, and RLC Circuits
Inductance, Inductors, RL Circuits & RC Circuits, LC, and RLC Circuits Self-inductance A time-varying current in a circuit produces an induced emf opposing the emf that initially set up the timevarying
More informationFrequency Bands. ω the numeric value of G ( ω ) depends on the frequency ω of the basis
1/28/2011 Frequency Bands lecture 1/9 Frequency Bands The Eigen value G ( ω ) of a linear operator is of course dependent on frequency ω the numeric value of G ( ω ) depends on the frequency ω of the basis
More informationYell if you have any questions
Class 31: Outline Hour 1: Concept Review / Overview PRS Questions possible exam questions Hour : Sample Exam Yell if you have any questions P31 1 Exam 3 Topics Faraday s Law Self Inductance Energy Stored
More informationChapter 10: Sinusoids and Phasors
Chapter 10: Sinusoids and Phasors 1. Motivation 2. Sinusoid Features 3. Phasors 4. Phasor Relationships for Circuit Elements 5. Impedance and Admittance 6. Kirchhoff s Laws in the Frequency Domain 7. Impedance
More information