CSE 245 ComputerAided Circuit Simulation and Verification Lecture

CSE 245: Computer-Aided Circuit Simulation and Verification Lecture 1: Introduction and Formulation Spring 2008 Chung-Kuan Cheng

Administration • CK Cheng, CSE 2130, tel. 534 -6184, ckcheng@ucsd. edu • Lectures: 12: 30 ~ 1: 50 pm TTH WLH 2205 • Textbooks – Electronic Circuit and System Simulation Methods T. L. Pillage, R. A. Rohrer, C. Visweswariah, Mc. Graw-Hill – Interconnect Analysis and Synthesis CK Cheng, J. Lillis, S. Lin, N. Chang, John Wiley & Sons • Grading – Homework and Projects: 60% – Project Presentation: 20% – Final Report: 20%

CSE 245: Course Outline • Formulation (2 -3 lectures) – – RLC Linear, Nonlinear Components, Transistors, Diodes Incident Matrix Nodal Analysis, Modified Nodal Analysis K Matrix • Linear System (3 -4 lectures) – – S domain analysis, Impulse Response Taylor’s expansion Moments, Passivity, Stability, Realizability Symbolic analysis, Y-Delta, BDD analysis • Matrix Solver (3 -4 lectures) – LU, KLU, reordering – Mutigrid, PCG, GMRES

CSE 245: Course Outline (Cont’) • Integration (3 -4 lectures) – – – Forward Euler, Backward Euler, Trapezoidal Rule Explicit and Implicit Method, Prediction and Correction Equivalent Circuit Errors: Local error, Local Truncation Error, Global Error A-Stable Alternating Direction Implicit Method • Nonlinear System (2 -3 lectures) – Newton Raphson, Line Search • Transmission Line, S-Parameter (2 -3 lectures) – FDTD: equivalent circuit, convolution – Frequency dependent components • Sensitivity • Mechanical, Thermal, Bio Analysis

Motivation • 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 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 • Derive from KCL/KVL • Sparse Tableau Analysis (IBM) • Nodal Analysis, Modified Nodal Analysis (SPICE) *some slides borrowed from Berkeley EE 219 Course

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

Branch Constitutive Equations (BCE) Ideal elements Element Branch Eqn Variable parameter Resistor v = R·i - Capacitor i = C·dv/dt - Inductor v = L·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 = A I · ic v=?

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 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 (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) 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 = A T e A K v. A T e = A i s

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 N+ N- N+ NN- Rkvalue 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: NC+ - N+ N- NC+ NC- Gkvalue N+ + vc NC- Gk NC+ N+ Gkvc N- NKCL at node N+ KCL at node N- NC-

Nodal Analysis – Current source “Stamp” Spice input format: Ik N+ N+ NN+ Ik N- N- N+ N- Ikvalue

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

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 + Ek N+ N- ik N+ N- Ekvalue 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: 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

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