CSE 291 Numerical Methods Lecture 1 Introduction and
- Slides: 48
CSE 291: Numerical Methods Lecture 1: Introduction and Formulation Spring 2020 Chung-Kuan Cheng
Administration • CK Cheng, CSE 2130, tel. 858 534 -6184, ckcheng+291@ucsd. edu • Lectures: 330 -450 PM TTH, Zoom • Grading – Project Proposal: 20% – Project Presentation: 40% – Final Report: 40%
References • 1. Electronic Circuit and System Simulation Methods, T. L. Pillage, R. A. Rohrer, C. Visweswariah, Mc. Graw-Hill, 1998 • 2. Convex Optimization Algorithms, D. P. Bertsekas, Athena Scientific 2015
CSE 245: Course Outline 1. Introduction 2. Problem Formulations 3. Integration Methods: matrix solvers, explicit and implicit integrations, matrix exponential methods, stability 4. System Solvers: 1) 2) 3) 4) Distributed computation: Parareal and multigrid in time Gradient descent methods Subgradient descent methods Homotopy methods 5. Sensitivity Analysis: direct method, adjoint network approach 6. Multiple Dimensional Analysis: Tensor decomposition 7. Source Localization
Motivation: Analysis and Optimization • Energy: Fission, Fusion, Fossil Energy, Efficiency Optimization • Astrophysics: Dark energy, Nucleosynthesis • Climate: Pollution, Weather Prediction • Biology: Microbial life • Socioeconomic Modeling: Global scale modeling Nonlinear Systems, ODE, PDE, Heterogeneous Systems, Multiscale Analysis.
Circuit Simulation: Overview stimulant generation netlist extraction, modeling Circuit Input and setup Simulator: Solve Complexity Accuracy Debug numerically Output frequency & time domain simulation user interface: worst cases, eye diagrams, noises
Circuit Simulation Circuit Input and setup Simulator: Solve numerically Output Types of analysis: – DC Analysis – DC Transfer curves – Transient Analysis – AC Analysis, Noise, Distortions, Sensitivity
Program Structure (a closer look) Models Input and setup Numerical Techniques: – Formulation of circuit equations – Solution of ordinary differential equations – Solution of nonlinear equations – Solution of linear equations Output
Lecture 1: Formulation • • Basic Elements KCL/KVL and Topology Sparse Tableau Analysis (IBM) Nodal Analysis, Modified Nodal Analysis (SPICE) *some slides borrowed from Berkeley EE 219 Course
Basic Elements • Two terminal elements • Multiple port elements Resistors: i=i(v) or v=v(i), e. g. i=v/R Capacitors: q=q(v) Inductors: �� = �� (i) Sources:
Basic Elements: capacitors •
Basic Elements: capacitors •
Basic Elements: capacitors •
Basic Elements: inductors •
Basic Elements: inductors •
Basic Elements: inductors •
Basic Elements: inductors •
Basic Elements: Summary For the simulation of two terminal elements, we can convert capacitors and inductors to resistors via Euler or trapezoidal integration. The conversion leaves two variables and one constraint. When the element is nonlinear, we need to watch out the slope of the device and the conservation of the charge or flux. Use examples of nonlinear and time varying capacitors to illustrate the formula of charge conservation.
Branch Constitutive Equations (BCE) Ideal elements Element Branch Eqn Variable parameter Resistor v = R·i v, i Capacitor i = C·dv/dt, i Inductor v = L·di/dt v, di/dt Voltage Source v = vs i = ? Current Source i = is v = ? VCVS vs = AV · vc i = ? VCCS is = GT · vc v = ? CCVS vs = RT · ic i = ? CCCS is = AI · ic v = ?
Conservation Laws • Determined by the topology of the circuit • Kirchhoff’s Current Law (KCL): The algebraic sum of all the currents flowing out of (or into) any circuit node is zero. – No Current Source Cut • Kirchhoff’s Voltage Law (KVL): Every circuit node has a unique voltage with respect to the reference node. The voltage across a branch vb is equal to the difference between the positive and negative referenced voltages of the nodes on which it is incident – No voltage source loop
Conservation Laws: Topology A circuit (V, E) can be decomposed into a spanning tree and links. The tree has n-1 (n=|V|) trunks, and m-n+1 (m=|E|) links. • A spanning tree that spans the nodes of the circuit. – Trunk of the tree: voltages of the trunks are independent. Check the case that the spanning tree does not exist! • Link that forms a loop with tree trunks – Link: currents of the links are independent. Check the case that the link does not exist! Thus, a circuit can be represented by n-1 trunk voltage and m-n+1 link currents.
Formulation of Circuit Equations • Unknowns – B branch currents (i) – N node voltages (e) – B branch voltages (v) • Equations – N+B Conservation Laws – B Constitutive Equations • 2 B+N equations, 2 B+N unknowns => unique solution
Equation Formulation - KCL R 3 1 R 1 2 R 4 G 2 v 3 0 Law: State Equation: Ai=0 N equations Node 1: Node 2: Branches Kirchhoff’s Current Law (KCL) Is 5
Equation Formulation - KVL R 3 1 R 1 2 Is 5 R 4 G 2 v 3 0 Law: State Equation: v - AT e = 0 B equations vi = voltage across branch i ei = voltage at node i Kirchhoff’s Voltage Law (KVL)
Equation Formulation - BCE R 3 1 R 1 Law: K vv + K i i = i s B equations 2 R 4 G 2 v 3 State Equation: 0 Is 5
Equation Formulation Node-Branch Incidence Matrix A branches n o 1 d 2 e s i 1 2 3 j B (+1, -1, 0) N { Aij = +1 if node i is + terminal of branch j -1 if node i is - terminal of branch j 0 if node i is not connected to branch j
Equation Assembly (Stamping Procedures) • Different ways of combining Conservation Laws and Branch Constitutive Equations – Sparse Table Analysis (STA) – Nodal Analysis (NA) – Modified Nodal Analysis (MNA)
Sparse Tableau Analysis (STA) 1. Write KCL: 2. Write KVL: 3. Write BCE: Ai=0 v - ATe=0 Kii + Kvv=S (N eqns) (B eqns) N+2 B eqns N+2 B unknowns N = # nodes B = # branches Sparse Tableau
Sparse Tableau Analysis (STA) Advantages • It can be applied to any circuit • Eqns can be assembled directly from input data • Coefficient Matrix is very sparse Disadvantages • Sophisticated programming techniques and data structures are required for time and memory efficiency
Nodal Analysis (NA) Use vector e as the only variables. Assume no voltage source 1. Write KCL Ai=0 (N equations, B unknowns) 2. Use BCE to relate branch currents to branch voltages i=f(v) (B equations B unknowns) 3. Use KVL to relate branch voltages to node voltages v=h(e) (B equations N unknowns) Yne=ins Nodal Matrix N eqns N unknowns N = # nodes
Nodal Analysis - Example R 3 1 R 1 1. KCL: 2. BCE: 3. KVL: Yne = ins Yn = AKv. AT Ins = Ais G 2 v 3 2 R 4 Is 5 0 Ai=0 Kvv + i = is - Kvv A Kvv = A is v = ATe A Kv. ATe = A is
Nodal Analysis • Example shows how NA may be derived from STA • Better Method: Yn may be obtained by direct inspection (stamping procedure) – Each element has an associated stamp – Yn is the composition of all the elements’ stamps
Nodal Analysis – Resistor “Stamp” Spice input format: Rk N+ N- Rkvalue N+ Rk N- N+ i N+ N- N- What if a resistor is connected to ground? …. Only contributes to the diagonal KCL at node N+ KCL at node N-
Nodal Analysis – VCCS “Stamp” Spice input format: Gk N+ N- NC+ NC- Gkvalue NC+ vc NC- N+ + - NC+ N+ Gkvc N- NKCL at node N+ KCL at node N- NC-
Nodal Analysis – Current source “Stamp” Spice input format: Ik N+ N- Ikvalue N+ N+ NN+ Ik N- N-
Nodal Analysis (NA) Advantages • Yn is often diagonally dominant and symmetric • Eqns can be assembled directly from input data • Yn has non-zero diagonal entries • Yn is sparse (not as sparse as STA) and smaller than STA: Nx. N compared to (N+2 B)x(N+2 B) Limitations • Conserved quantity must be a function of node variable – Cannot handle floating voltage sources, VCVS, CCCS, CCVS. How do we handle the current variable? – Hint: No cut of currents
Modified Nodal Analysis (MNA) How do we deal with independent voltage sources? + Ekl l k ikl k l • ikl cannot be explicitly expressed in terms of node voltages it has to be added as unknown (new column) • ek and el are not independent variables anymore a constraint has to be added (new row)
MNA – Voltage Source “Stamp” Spice input format: Vk N+ N- Ekvalue + Ek N+ N- ik N+ N- ik N+ 0 0 1 N- 0 0 -1 Branch k 1 -1 0 RHS
Modified Nodal Analysis (MNA) How do we deal with independent voltage sources? Augmented nodal matrix In general B= -CT Why? Some branch currents
MNA – General rules • A branch current is always introduced as an additional variable for a voltage source or an inductor • For current sources, resistors, conductors and capacitors, the branch current is introduced only if: – Any circuit element depends on that branch current – That branch current is requested as output
MNA – CCCS and CCVS “Stamp”
MNA – An example 1 R 1 + v 3 R 3 2 Is 5 R 4 G 2 v 3 0 - ES 6 + E 7 v 3 Step 1: Write KCL (1) (2) (3) (4) 3 + R 8 4
MNA – An example Step 2: Use branch equations to eliminate as many branch currents as possible (1) (2) (3) (4) Step 3: Write down unused branch equations (b 6) (b 7)
MNA – An example Step 4: Use KVL to eliminate branch voltages from previous equations (1) (2) (3) (4) (b 6) (b 7)
MNA – An example
MNA – An example 1 R 1 + v 3 R 3 2 R 4 G 2 v 3 0 Is 5 - ES 6 3 + + E 7 v 3 R 8 4
Modified Nodal Analysis (MNA) Advantages • MNA can be applied to any circuit • Eqns can be assembled directly from input data • MNA matrix is close to Yn Limitations • Sometimes we have zeros on the main diagonal
Formulation: Summary Ingredients • Basic Elements • Topology: Tree trunks and links Formats • STA, NA, MNA Are there other formats? • Use an example of link analysis.
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