Transcript Lecture 1
CSE245: Computer-Aided Circuit Simulation and Verification Lecture 1: Introduction and Formulation Winter 2015 Chung-Kuan Cheng Administration • CK Cheng, CSE 2130, tel. 858 534-6184, [email protected] • Lectures: 5:00 ~ 6:20pm TTH CSE2217, • Grading – Homework: 60% – Project Presentation: 20% – Final Report: 20% References • 1. Electronic Circuit and System Simulation Methods, T.L. Pillage, R.A. Rohrer, C. Visweswariah, McGraw-Hill, 1998 • 2. Interconnect Analysis and Synthesis, CK Cheng, J. Lillis, S.Lin and N. Chang, John Wiley, 2000 • 3. Computer-Aided Analysis of Electronic Circuits, L.O. Chua and P.M. Lin, Prentice Hall, 1975 • 4. A Friendly Introduction to Numerical Analysis, B. Bradie, Pearson/Prentice Hall, 2005, http://www.pcs.cnu.edu/~bbradie/textbookanswers.html • 5. Numerical Recipes: The Art of Scientific Computing, Third Edition, W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Cambridge University Press, 2007. • 6. Numerical Algorithms, Justin Solomon. CSE245: Course Outline 1. Introduction 2. Problem Formulations: basic elements, circuit topology, network regularization 3. Linear Circuits: matrix solvers, explicit and implicit integrations, matrix exponential methods, convergence 4. Nonlinear Systems: Newton-Raphson method, Nesterov methods, homotopy methods 5. Sensitivity Analysis: direct method, adjoint network approach 6. Various Simulation Approaches: FDM, FEM, BEM, FFT, multipole methods, Monte Carlo, random walks 7. Multiple Dimensional Analysis: Tensor decomposition 8. Applications: power distribution networks, IO circuits, full wave analysis Motivation: Analysis • 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. Motivation: Analysis Exascale Calculation: 1018, Parallel Processing at 20MW • 10 Petaflops 2012 • 100 PF 2017 • 1000 PF 2022 Motivation: Analysis • Modeling: Inputs, outputs, system models. • Simulation: Time domain, frequency domain, wavelet simulation. • Sensitivity Calculation: Optimization • Uncertainty Quantification: Derivation with partial information or variations • User Interface: Data mining, visualization. Motivation: Circuit Analysis • Why – Whole Circuit Analysis, Interconnect Dominance • What – Power, Clock, Interconnect Coupling • Where – – – – Matrix Solvers, Integration Methods RLC Reduction, Transmission Lines, S Parameters Parallel Processing Thermal, Mechanical, Biological 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 f (X ) C dX (t ) dt Output numerically dX (t ) f (X ) C GX (t ) BU (t ) dt Y DX (t ) FU (t ) 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 EE219 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: 𝝓= 𝝓(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 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 • 2B+N equations, 2B+N unknowns => unique solution Equation Formulation - KCL R3 1 R1 2 R4 G2v3 0 Law: State Equation: Ai=0 N equations Node 1: Node 2: i1 i 2 1 1 1 0 0 0 0 0 1 1 1 i3 0 i4 Branches i5 Kirchhoff’s Current Law (KCL) Is5 Equation Formulation - KVL R3 1 R1 2 Is5 R4 G2v3 0 Law: v - AT State Equation: e=0 B equations v1 1 0 0 v 1 0 0 2 e v3 1 1 1 0 e2 v4 0 1 0 v5 0 1 0 vi = voltage across branch i ei = voltage at node i Kirchhoff’s Voltage Law (KVL) Equation Formulation - BCE R3 1 R1 Law: K v v + K ii = i s B equations 2 R4 G2v3 State Equation: 1 R 1 0 0 0 0 0 0 0 0 0 0 0 0 0 G2 1 0 R3 0 0 Is5 1 R4 0 0 v i 0 1 1 0 v i2 0 2 0 v3 i3 0 v4 i4 0 0 v i i 5 5 s5 0 Equation Formulation Node-Branch Incidence Matrix A branches 1 2 3 n o 1 d 2 e s i j (+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 B 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 0 i 0 A 0 0 I AT v 0 K i K v 0 e S Sparse Tableau (N eqns) (B eqns) (B eqns) N+2B eqns N+2B unknowns N = # nodes B = # branches 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 R3 1 R1 1. KCL: 2. BCE: 3. KVL: Yne = ins Yn = AKvAT Ins = Ais 2 Is5 R4 G2v3 0 Ai=0 Kvv + i = is i = is - Kvv A Kvv = A is v = ATe A KvATe = A is 1 1 G 2 R R3 1 1 R3 1 R3 e1 0 1 1 e2 is 5 R3 R4 G2 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: N+ Rk N- Rk N+ i 1 N+ R k 1 N- R k N+ NN- 1 Rk 1 Rk Rkvalue What if a resistor is connected to ground? …. Only contributes to the diagonal 1 iothers R eN eN is k KCL at node N+ 1 iothers R eN eN is k KCL at node N- Nodal Analysis – VCCS “Stamp” Spice input format: NC+ i i others others N+ N- NC+ NC- Gkvalue N+ + vc NC- Gk NC+ N+ G k G k N- Gkvc Gk eNC eNC is Gk eNC eNC is NKCL at node N+ KCL at node N- NC- Gk Gk Nodal Analysis – Current source “Stamp” Spice input format: Ik N+ N- Ikvalue N+ N+ N- N+ Ik N- N- I k I k 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: NxN compared to (N+2B)x(N+2B) 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 1 1 1 ek 1 el 0 ikl Ekl • 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 + Ek N+ - Nik N+ N- Ekvalue N+ N- ik N+ 0 0 1 N- 0 0 -1 Branch k 1 -1 0 RHS 0 0 Ek Modified Nodal Analysis (MNA) How do we deal with independent voltage sources? Augmented nodal matrix Yn C B e MS 0 i Some branch currents In general B= -CT Why? Yn C B e MS D i 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 R1 + v3 R3 2 - R8 - + 0 4 E7v3 Step 1: Write KCL i1 i2 i3 0 i3 i4 i5 i6 0 i6 i8 0 i7 i8 8 3 + Is5 R4 G2v3 ES6 (1) (2) (3) (4) MNA – An example Step 2: Use branch equations to eliminate as many branch currents as possible 1 1 v1 G2v3 v3 0 R1 R3 (1) 1 1 v3 v4 i6 is 5 R3 R4 (2) i6 1 v8 0 R8 (3) i7 1 v8 0 R8 (4) Step 3: Write down unused branch equations v6 ES6 v7 E7 v3 0 (b6) (b7) MNA – An example Step 4: Use KVL to eliminate branch voltages from previous equations 1 1 e1 G2 (e1 e2 ) (e1 e2 ) 0 R1 R3 1 1 (e1 e2 ) e 2 i6 i s 5 R3 R4 (1) (2) i6 1 ( e3 e 4 ) 0 R8 (3) i7 1 ( e3 e 4 ) 0 R8 (4) (e3 e2 ) ES 6 e4 E 7 (e1 e2 ) 0 (b6) (b7) MNA – An example Yn C B e MS 0 i 1 1 G2 R3 R1 1 R3 0 0 0 E7 1 G2 R3 1 1 R3 R4 0 0 0 0 1 1 R8 1 R8 1 E7 0 0 0 1 R8 1 R8 0 1 0 0 e 0 1 1 0 e i 2 s5 e 0 3 1 0 e4 0 0 1 i6 ES 6 i7 0 0 0 0 0 MNA – An example + v3 R3 1 R1 2 Is5 R4 G2v3 ES6 - 3 + + R8 0 1 1 G 2 R R3 1 1 R3 0 0 0 E7 1 G2 R3 1 1 R3 R4 4 Ev 0 0 0 0 1 1 R8 1 R8 1 E7 0 0 0 1 R8 1 R8 0 1 7 3 0 0 e 0 1 1 0 e i 2 s 5 e 0 3 1 0 e4 0 0 1 i6 ES 6 i7 0 0 0 0 0 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.