ee 616 computer aided analysis of electronic networks lecture 12
DESCRIPTION
EE 616 Computer Aided Analysis of Electronic Networks Lecture 12. Instructor: Dr. J. A. Starzyk, Professor School of EECS Ohio University Athens, OH, 45701. Note: materials in this lecture are from the notes of EE219A UC-berkeley http://www- cad.eecs.berkeley.edu/~nardi/EE219A/contents.html. - PowerPoint PPT PresentationTRANSCRIPT
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EE 616 Computer Aided Analysis of Electronic Networks
Lecture 12
Instructor: Dr. J. A. Starzyk, ProfessorSchool of EECSOhio UniversityAthens, OH, 45701
Note: materials in this lecture are from the notes of EE219A UC-berkeleyhttp://www- cad.eecs.berkeley.edu/~nardi/EE219A/contents.html
2
Outline Transient Analysis of dynamical circuits
– i.e., circuits containing C and/or L Examples Solution of Ordinary Differential Equations (Initial Value
Problems – IVP)– Forward Euler (FE), Backward Euler (BE) and
Trapezoidal Rule (TR)– Multistep methods– Convergence
Methods for Ordinary Differential Equations
By Prof. Alessandra Nardi
3
Ground Plane
Signal Wire
LogicGate
LogicGate
• Metal Wires carry signals from gate to gate.• How long is the signal delayed?
Wire and ground plane form a capacitor
Wire has resistance
Application Problems
Signal Transmission in an Integrated Circuit
4
capacitor
resistor
• Model wire resistance with resistors.• Model wire-plane capacitance with capacitors.
Constructing the Model• Cut the wire into sections.
Application Problems
Signal Transmission in an IC – Circuit Model
5
Nodal Equations Yields 2x2 System
C1
R2
R1 R3 C2
Constitutive Equations
cc
dvi C
dt
1R Ri v
R
Conservation Laws
1 1 20C R Ri i i
2 3 20C R Ri i i
1
1 2 21 1
2 22
2 3 2
1 1 1
0
0 1 1 1
dvR R RC vdt
C vdv
R R Rdt
1Ri
1Ci
2Ri
2Ci
3Ri
1v 2v
Application Problems
Signal Transmission in an IC – 2x2 example
6eigenvectors
1
1 2 21 1
2 22
2 3 2
1 1 1
0
0 1 1 1
dvR R RC vdt
C vdv
R R Rdt
1 2 1 3 2Let 1, 10, 1C C R R R 1.1 1.0
1.0 1.1
A
dxx
dt
11 1 0.1 0 1 1
1 1 0 2.1 1 1A
Eigenvalues and Eigenvectors
Eigenvalues
Application Problems
Signal Transmission in an IC – 2x2 example
1Change of variab (le ) ( ) ( ) (s ): Ey t x t y t E x t
1
1 2 1 2
10 0
0 0
0 0n n
n
A E E E E E
E
E
Eigendecomposition:
0
( )Substituting: ( ), (0)
dEy tAEy t Ey x
dt
11Multiply by ( ) : ( )dy t
E AE tdt
E y 0 0
10 0
0 0
( )n
y t
0Consider an ODE( )
( ), (0): dx t
Ax t x xdt
An Aside on Eigenanalysis
8
( )Decoupling: ( ) ( ) (0)iti
i i i
dy ty t y t e y
dt
1 0 0
0 0
0 0
From last slide:( ) ( )
n
dy tdt
y t
DecoupledEquations!
1) Determine , E 1 0 0
3) Compute ( ) 0 0 (0)
0 0 n
t
t
e
y t y
e
102) Compute (0) y E x
4) ( ) ( ) x t Ey t
0Steps for solvi( )
( ), (0)ng dx t
Ax t x xdt
An Aside on Eigenanalysis
9
1(0) 1v
2 (0) 0v
Notice two time scale behavior
• v1 and v2 come together quickly (fast eigenmode).• v1 and v2 decay to zero slowly (slow eigenmode).
Application Problems
Signal Transmission in an IC – 2x2 example
10
Circuit Equation Formulation
For dynamical circuits the Sparse Tableau equations can be written compactly:
For sake of simplicity, we shall discuss first order ODEs in the form:
riablescircuit va of vector theis where
)0(
0),,)(
(
0
x
xx
txdt
tdxF
),()(
txfdt
tdx
11
Ordinary Differential Equations
Initial Value Problems (IVP)
Typically analytic solutions are not available
solve it numerically
.condition initial given the intervalan in
)(
),()(
:(IVP) Problem Value Initial Solve
00
00
x,T][t
xtx
txfdt
tdx
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Not necessarily a solution exists and is unique for:
It turns out that, under rather mild conditions on the continuity and differentiability of F, it can be proven that there exists a unique solution.
Also, for sake of simplicity only consider
linear case:
0),,( tydt
dyF
We shall assume that has a unique solution 0),,( tydt
dyF
00 )(
)()(
xtx
tAxdt
tdx
Ordinary Differential Equations Assumptions and Simplifications
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First - Discretize Time
Second - Represent x(t) using values at ti
ˆ ( )llx x t
Approx. sol’n
Exact sol’n
Third - Approximate using the discrete ( )ldx t
dtˆ 'slx
1 1
1
ˆ ˆ ˆ ˆExample: ( )
l l l l
ll l
d x x x xx t or
dt t t
Lt T1t 2t 1Lt 0
t t t
1t 2t 3t Lt0
3x̂ 4x̂1x̂
2x̂
Finite Difference Methods
Basic Concepts
14
lt 1lt t
x
1( ) ( )slope l lx t x t
t
slope ( )ldx t
dt
1( ) ( ) ( )l l lx t x t t A x t
1
1
( ) ( )( ) ( )
or
( ) ( ) ( )
l ll l
l l l
x t x tdx t A x t
dt t
x t x t t A x t
Finite Difference Methods
Forward Euler Approximation
15
1t 2t t
x
(0)tAx
11 ˆ( ) (0) 0x t x x tAx
3t
2 1 12 ˆ ˆ ˆ( )x t x x tAx
1ˆtAx
1 1ˆ ˆ ˆ( ) L L LLx t x x tAx
Finite Difference Methods
Forward Euler Algorithm
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lt 1lt t
x 1slope ( )l
dx t
dt
1( ) ( )slope l lx t x t
t
1 1( ) ( ) ( )l l lx t x t t A x t
11 1
1 1
( ) ( )( ) ( )
or
( ) ( ) ( )
l ll l
l l l
x t x tdx t A x t
dt t
x t x t t A x t
Finite Difference Methods
Backward Euler Approximation
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1t 2t t
x
1ˆtAx
2ˆtAx
1 11 ˆ ˆ( ) (0)x t x x tAx
Solve with Gaussian Elimination
1ˆ[ ] (0)I tA x x
1 1ˆ ˆ( ) [ ]L LLx t x I tA x
2 1 12 ˆ ˆ( ) [ ]x t x I tA x
Finite Difference Methods
Backward Euler Algorithm
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1
1
1
1 1
1( ( ) ( ))
21
( ( ) ( ))2
( ) ( )
1( ) ( ) ( ( ) ( ))
2
l l
l l
l l
l l l l
d dx t x t
dt dt
Ax t Ax t
x t x t
t
x t x t tA x t x t
t
x
1( ) ( )slope l lx t x t
t
slope ( )ldx t
dt
1slope ( )l
dx t
dt
1 1
1 1( ( ) ( )) ( ( ) ( ))
2 2l l l lx t tAx t x t tAx t
Finite Difference Methods
Trapezoidal Rule Approximation
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1t 2t t
x1ˆ (0)
2 2
t tI A x I A x
Solve with Gaussian Elimination
1 11 ˆ ˆ( ) (0) (0)
2
tx t x x Ax Ax
12 1
2
11
ˆ ˆ( )2 2
ˆ ˆ( )2 2
L LL
t tx t x I A I A x
t tx t x I A I A x
Finite Difference Methods
Trapezoidal Rule Algorithm
20
1
1( ) (( ) )) )( (l
l
t
l l t
dx t x t x t A dx t xA
dt
lt 1lt
1
( )l
l
t
tAx d
1( )ltAx t BE
( )ltAx t FE
( ) ( )2 l l
tAx t Ax t
Trap
Finite Difference Methods
Numerical Integration View
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Finite Difference Methods - Sources of Error
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Trap Rule, Forward-Euler, Backward-Euler Are all one-step methods
Forward-Euler is simplest No equation solution explicit method. Box approximation to integral
Backward-Euler is more expensive Equation solution each step implicit method
Trapezoidal Rule might be more accurate Equation solution each step implicit method Trapezoidal approximation to integral
1 2 3ˆ ˆ ˆ ˆ is computed using only , not , , etc.l l l lx x x x
Finite Difference Methods
Summary of Basic Concepts
23
( ) ( ( ), ( ))dx t f x t u t
dtNonlinear Differential Equation:
k-Step Multistep Approach: 0 0
ˆ ˆ ,k k
l j l jj j l j
j j
x t f x u t
Solution at discrete points
Time discretization
Multistep coefficients
2ˆ lx
lt1lt 2lt 3lt l kt
ˆ lx1ˆ lx
ˆ l kx
Multistep Methods
Basic Equations
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Multistep Equation:
1 1 1,l l l lx t x t t f x t u t
FE Discrete Equation: 1 11ˆ ˆ ˆ ,l l l
lx x t f x u t
0 1 0 11, 1, 1, 0, 1k
Forward-Euler Approximation:
Multistep Coefficients:
Multistep Coefficients:
BE Discrete Equation:
0 1 0 11, 1, 1, 1, 0k 1ˆ ˆ ˆ ,l l l
lx x t f x u t
Trap Discrete Equation: 1 11ˆ ˆ ˆ ˆ, ,
2l l l l
l l
tx x f x u t f x u t
0 1 0 1
1 11, 1, 1, ,
2 2k Multistep Coefficients:
0 0
ˆ ˆ ,k k
l j l jj j l j
j j
x t f x u t
Multistep Methods – Common Algorithms
TR, BE, FE are one-step methods
25
Multistep Equation:
01) If 0 the multistep method is implicit 2) A step multistep method uses previous ' and 'k k x s f s
03) A normalization is needed, 1 is common 4) A -step method has 2 1 free coefficientsk k
How does one pick good coefficients?
Want the highest accuracy
0 0
ˆ ˆ ,k k
l j l jj j l j
j j
x t f x u t
Multistep Methods
Definition and Observations
26
Definition: A finite-difference method for solving initial value problems on [0,T] is said to be convergent if given any A and any initial condition
0,
ˆmax 0 as t 0lT
lt
x x l t
exactx
tˆ computed with
2lx
ˆ computed with tlx
Multistep Methods – Convergence Analysis
Convergence Definition
27
Definition: A multi-step method for solving initial value problems on [0,T] is said to be order p convergent if given any A and any initial condition
0,
ˆmaxpl
Tl
t
x x l t C t
0for all less than a given t t
Forward- and Backward-Euler are order 1 convergent
Trapezoidal Rule is order 2 convergent
Multistep Methods – Convergence Analysis
Order-p Convergence
28
Multistep Methods – Convergence Analysis
Two types of error
made.been haserror previous no assuming
,solution theof eexact valu theand ˆ value
computed ebetween th difference theis at method
n integratioan of (LTE)Error Truncation Local The
11
1
)x(tx
t
ll
l
exactly.known iscondition initial only the that assuming
,solution theof eexact valu theand ˆ value
computed ebetween th difference theis at method
n integratioan of (GTE)Error Truncation Global The
11
1
)x(tx
t
ll
l
29
For convergence we need to look at max error over the whole time interval [0,T]– We look at GTE
Not enough to look at LTE, in fact:– As I take smaller and smaller time steps t,
I would like my solution to approach exact
solution better and better over the whole time interval, even though I have to add up LTE
from more time steps.
Multistep Methods – Convergence Analysis
Two conditions for Convergence
30
1) Local Condition: One step errors are small (consistency)
2) Global Condition: The single step errors do not grow too quickly (stability)
Typically verified using Taylor Series
All one-step methods are stable in this sense.
Multistep Methods – Convergence Analysis
Two conditions for Convergence
31
Definition: A one-step method for solving initial value problems on an interval [0,T] is said to be consistent if for any A and any initial condition
1ˆ0 as t 0
x x t
t
One-step Methods – Convergence Analysis
Consistency definition
32
One-step Methods – Convergence Analysis
Consistency for Forward Euler
33
One-step Methods – Convergence Analysis
Convergence Analysis for Forward Euler
Forward-Euler definition
Expanding in t about yields
where is the "one-step" error bounded by
34
One-step Methods – Convergence Analysis
Convergence Analysis for Forward Euler
35
One-step Methods – Convergence Analysis
A helpful bound on difference equations
36
One-step Methods – Convergence Analysis
A helpful bound on difference equations
37
One-step Methods – Convergence Analysis
Back to Convergence Analysis for Forward Euler
38
• Forward-Euler is order 1 convergent• The bound grows exponentially with time interval• C is related to the solution second derivative• The bound grows exponentially fast with norm A.
One-step Methods – Convergence Analysis
Observations about Convergence Analysis for FE
39
Summary
Transient Analysis of dynamical circuits– i.e., circuits containing C and/or L
Examples Solution of Ordinary Differential Equations
(Initial Value Problems – IVP)– Forward Euler (FE), Backward Euler (BE) and
Trapezoidal Rule (TR)– Multistep methods– Convergence
40
Multistep Methods - Local Truncation Error
41
Local Truncation Error (cont’d)
42
Examples
43
Examples (cont’d)
44
Determination of Local Error
45
Implicit Methods
46
Convergence
47
Convergence (cont’d)
48
Convergence (cont’d)
49
Other methods
50
Summary