CSE 245: Computer Aided Circuit Simulation and Verification Instructor: Prof. Chung-Kuan Cheng Winter 2003 Lecture 1: Formulation

Jan. 24, 2003Cheng & Zhu, Agenda  RCL Network  Sparse Tableau Analysis  Modified Nodal Analysis

Jan. 24, 2003Cheng & Zhu, History of SPICE  SPICE -- Simulation Program with Integrated Circuit Emphasis  1969, CANCER developed by Laurence Nagel on Prof. Ron Roher ’ s class  1970~1972, CANCER program  May 1972, SPICE-I release  July ’ 75, SPICE 2A, …, 2G  Aug 1982, SPICE 3 (in C language)  No new progress on software package since then

Jan. 24, 2003Cheng & Zhu, RCL circuit

Jan. 24, 2003Cheng & Zhu, RCL circuit (II)  General Circuit Equation  Consider homogeneous form first Q: How to Compute A k ? and

Jan. 24, 2003Cheng & Zhu,  Assume A has non-degenerate eigenvalues and corresponding linearly independent eigenvectors, then A can be decomposed as where and Solving RCL Equation

Jan. 24, 2003Cheng & Zhu,  What ’ s the implication then?  To compute the eigenvalues: Solving RCL Equation (II) real eigenvalue Conjugative Complex eigenvalue where

Jan. 24, 2003Cheng & Zhu, Solving RCL Equation (III) In the previous example where hence Let c=r=l=1, we have

Jan. 24, 2003Cheng & Zhu,  What if matrix A has degenerated eigenvalues? Jordan decomposition ! Solving RCL Equation (IV) J is in the Jordan Canonical form And still

Jan. 24, 2003Cheng & Zhu, Jordan Decomposition similarly

Jan. 24, 2003Cheng & Zhu, Agenda  RCL Network  Sparse Tableau Analysis  Modified Nodal Analysis

Jan. 24, 2003Cheng & Zhu, Equation Formulation  KCL Converge of node current  KVL Closure of loop voltage  Brach equations I, R relations

Jan. 24, 2003Cheng & Zhu, Types of elements  Resistor  Capacitor  Inductor L is even dependent on frequency due to skin effect, etc …  Controlled Sources VCVS, VCCS, CCVS, CCCS

Jan. 24, 2003Cheng & Zhu, Cut-set analysis 1. Construct a spanning tree 2. Take as much capacitor branches as tree branches as possible 3. Derive the fundamental cut-set, in which each cut truncates exactly one tree branch 4. Write KCL equations for each cut 5. Write KVL equations for each tree link 6. Write the constitution equation for each branch

Jan. 24, 2003Cheng & Zhu, KCL Formulation #nodes-1 lines #braches columns

Jan. 24, 2003Cheng & Zhu, KCL Formulation (II)  Permute the columns to achieve a systematic form

Jan. 24, 2003Cheng & Zhu, KVL Formulation Remove the equations for tree braches and systemize

Jan. 24, 2003Cheng & Zhu, Cut & Loop relation In the previous example

Jan. 24, 2003Cheng & Zhu, Sparse Tableau Analysis (STA)  n=#nodes, b=#branches (n-1) KCL b KVL b branch relations b b n-1 Totally 2b+n-1 variables, 2b+n-1 equations bbn-1 Due to independent sources

Jan. 24, 2003Cheng & Zhu, STA (II)  Advantages Covers any circuit Easy to assemble Very sparse  K i, K v, I each has exactly b non-zeros. A and A T each has at most 2b non-zeros.  Disadvantages Sophisticated data structures & programming techniques

Jan. 24, 2003Cheng & Zhu, Agenda  RCL Network  Sparse Tableau Analysis  Modified Nodal Analysis

Jan. 24, 2003Cheng & Zhu, Nodal Analysis  Derivation From STA: (1) (2) (3) (3) x K i -1  (4) x A  (4) Using (a)  (5) (6) Tree trunk voltages Substitute with node voltages (to a given reference), we get the nodal analysis equations.

Jan. 24, 2003Cheng & Zhu, Nodal Analysis (II)

Jan. 24, 2003Cheng & Zhu, Modified Nodal Analysis  General Form Node Conductance matrix KCL Independent current source Independent voltage source Due to non-conductive elements  Y n can be easily derived  Add extra rows/columns for each non-conductive elements using templates

Jan. 24, 2003Cheng & Zhu, MNA (II)  Fill Y n matrix according to incidence matrix Choose n 6 as reference node

Jan. 24, 2003Cheng & Zhu, MNA Templates Independent current source Independent voltage source Add to the right-hand side of the equation

Jan. 24, 2003Cheng & Zhu, MNA Templates (II) CCVS CCCS

Jan. 24, 2003Cheng & Zhu, MNA Templates (III) VCVS VCCS

Jan. 24, 2003Cheng & Zhu, MNA Templates (IV) M Mutual inductance Operational Amplifier

Jan. 24, 2003Cheng & Zhu, MNA Example Circuit Topology MNA Equations

Jan. 24, 2003Cheng & Zhu, MNA Summary  Advantages Covers any circuits Can be assembled directly from input data. Matrix form is close to Y n  Disadvantages We may have zeros on the main diagonal. Principle minors could be singular