Formulation of Circuit Equations

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Formulation of Circuit Equations

• Lecture 2
• Alessandra Nardi

Thanks to Prof. Sangiovanni-Vincentelli and Prof.
Newton
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219A Course Overview

• Fundamentals of Circuit Simulation
• Approximately 12 lectures
• Analog Circuits Simulation 
• Approximately 4 lectures
• Digital Systems Verification 
• Approximately 3 lectures
• Physical Issues Verification 
• Approximately 6 lectures

E.g. SPICE, HSPICE, PSPICE, SPECTRE, ELDO .
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SPICE historyProf. Pederson with a cast of
thousands

• 1969-70 Prof. Roher and a class project
• CANCER Computer Analysis of Nonlinear Circuits,
  Excluding Radiation
• 1970-72 Prof. Roher and Nagel
• Develop CANCER into a truly public-domain,
  general-purpose circuit simulator
• 1972 SPICE I released as public domain
• SPICE Simulation Program with Integrated Circuit
  Emphasis
• 1975 Cohen following Nagel research
• SPICE 2A released as public domain
• 1976 SPICE 2D New MOS Models
• 1979 SPICE 2E Device Levels (R. Newton appears)
• 1980 SPICE 2G Pivoting (ASV appears)

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Circuit Simulation

• Types of analysis
• DC Analysis
• DC Transfer curves
• Transient Analysis
• AC Analysis, Noise, Distorsion, Sensitivity

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Program Structure (a closer look)
Input and setup
Models

• Numerical Techniques
• Formulation of circuit equations
• Solution of linear equations
• Solution of nonlinear equations
• Solution of ordinary differential equations

Output
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Formulation of Circuit Equations
Set of equations
Circuit with B branches N nodes
Simulator
Set of unknowns
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Formulation of Circuit Equations

• Unknowns
• B branch currents (i)
• N node voltages (e)
• B branch voltages (v)
• Equations
• NB Conservation Laws
• B Constitutive Equations

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Branch Constitutive Equations (BCE)

• Determined by the mathematical model of the
  electrical behavior of a component
• Example VRI
• In most of circuit simulators this mathematical
  model is expressed in terms of ideal elements

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Ideal Elements Reference Direction

• Branch voltages and currents are measured
  according to the associated reference directions
• Also define a reference node (ground)

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Branch Constitutive Equations (BCE)

• Ideal elements

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Conservation Laws

• Determined by the topology of the circuit
• Kirchhoffs Voltage Law (KVL) Every circuit node
  has a unique voltage with respect to the
  reference node. The voltage across a branch eb is
  equal to the difference between the positive and
  negative referenced voltages of the nodes on
  which it is incident
• Kirchhoffs Current Law (KCL) The algebraic sum
  of all the currents flowing out of (or into) any
  circuit node is zero.

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Equation Formulation - KCL
A i 0
N equations
Kirchhoffs Current Law (KCL)
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Equation Formulation - KVL
v - AT e 0
B equations
Kirchhoffs Voltage Law (KVL)
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Equation Formulation - BCE
Kvv i is
B equations
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Equation FormulationNode-Branch Incidence Matrix

• PROPERTIES
• A is unimodular
• 2 nonzero entries in each column

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Equation Assembly (Stamping Procedures)

• Different ways of combining Conservation Laws and
  Constitutive Equations
• Sparse Table Analysis (STA)
• Brayton, Gustavson, Hachtel
• Modified Nodal Analysis (MNA)
• McCalla, Nagel, Roher, Ruehli, Ho

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Sparse Tableau Analysis (STA)

• Write KCL Ai0 (N eqns)
• Write KVL v -ATe0 (B eqns)
• Write BCE Kii KvvS (B eqns)

N2B eqns N2B unknowns
N nodes B branches
Sparse Tableau
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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
• Problem
• Sophisticated programming techniques and data
• structures are required for time and memory
• efficiency

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Nodal Analysis (NA)

• 1. Write KCL
• Ai0 (N eqns, B unknowns)
• 2. Use BCE to relate branch currents to branch
  voltages
• if(v) (B unknowns ? B unknowns)
• Use KVL to relate branch voltages to node
  voltages
• vh(e) (B unknowns ? N unknowns)

N eqns N unknowns
Yneins
N nodes
Nodal Matrix
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Nodal Analysis - Example
R3

• KCL Ai0
• BCE Kvv i is ? i is - Kvv ? A Kvv A
  is
• KVL v ATe ? A KvATe A is

Yne ins
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Nodal Analysis

• Example shows NA may be derived from STA
• Better Yn may be obtained by direct inspection
  (stamping procedure)
• Each element has an associated stamp
• Yn is the composition of all the elements stamps

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Nodal Analysis Resistor Stamp
Spice input format Rk N N- Rkvalue
What if a resistor is connected to
ground? . Only contributes to the diagonal
KCL at node N KCL at node N-
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Nodal Analysis VCCS Stamp
Spice input format Gk N N- NC NC-
Gkvalue
vc -
KCL at node N KCL at node N-
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Nodal Analysis Current source Stamp
Spice input format Ik N N- Ikvalue
N N-
N N-
Ik
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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 (N2B)x(N2B)
• Limitations
• Conserved quantity must be a function of node
  variable
• Cannot handle floating voltage sources, VCVS,
  CCCS, CCVS

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Modified Nodal Analysis (MNA)
How do we deal with independent voltage sources?
Ekl
k l

-
k
l
ikl

• 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)

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MNA Voltage Source Stamp
Spice input format ESk N N- Ekvalue
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Modified Nodal Analysis (MNA)

• How do we deal with independent voltage sources?
• Augmented nodal matrix

In general
Some branch currents
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MNA General rules

• A branch current is always introduced as and
  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

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MNA CCCS and CCVS Stamp
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MNA An example
1
Is5
R4
G2v3
-

0
4
E7v3
Step 1 Write KCL i1 i2 i3 0 (1) -i3
i4 - i5 - i6 0 (2) i6 i8
0 (3) i7 i8 0 (4)
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MNA An example
Step 2 Use branch equations to eliminate as many
branch currents as possible 1/R1v1 G2 v3
1/R3v3 0 (1) - 1/R3v3 1/R4v4 - i6
is5 (2) i6 1/R8v8 0 (3) i7
1/R8v8 0 (4) Step 3 Write down unused
branch equations v6 ES6 (b6) v7 E7v3
0 (b7)
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MNA An example
Step 4 Use KVL to eliminate branch voltages from
previous equations 1/R1e1 G2(e1-e2)
1/R3(e1-e2) 0 (1) - 1/R3(e1-e2) 1/R4e2 -
i6 is5 (2) i6 1/R8(e3-e4) 0 (3) i7
1/R8(e3-e4) 0 (4) (e3-e2)
ES6 (b6) e4 E7(e1-e2) 0 (b7)
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MNA An example
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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 and
  principle minors may also be singular.