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CSE245: Computer-Aided Circuit Simulation and Verification. Lecture Note 5 Numerical Integration Spring 2010 Prof. Chung-Kuan Cheng. Numerical Integration: Outline. One-step Method for ODE (IVP) Forward Euler Backward Euler Trapezoidal Rule Equivalent Circuit Model Convergence Analysis - PowerPoint PPT Presentation
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CSE245: Computer-Aided Circuit Simulation and
VerificationLecture Note 5
Numerical Integration
Spring 2010
Prof. Chung-Kuan Cheng
1
Numerical Integration: Outline
• One-step Method for ODE (IVP)– Forward Euler– Backward Euler– Trapezoidal Rule– Equivalent Circuit Model
• Convergence Analysis
• Linear Multi-Step Method
• Time Step Control
2
Ordinary Difference Equaitons
.condition initial given the intervalan in
)(
),()(
:(IVP) Problem Value Initial Solve
00
00
x,T][t
xtx
txfdt
tdx
N equations, n x variables, n dx/dt.
Typically analytic solutions are not available
solve it numerically
3
Numerical Integration
Forward Euler
Backward Euler
Trapezoidal
0 0
( )( , )
( )
dx tf x t
dtx t x
4
Numerical Integration: State Equation
Forward Euler
Backward Euler
5
Numerical Integration: State Equation
Trapezoidal
6
7
Equivalent Circuit Model-BE• Capacitor
( ) ( ) ( )tCv t t v t i t t
+
C
-
+
-
( )v t t
( )i t t
( )i t t
Ceq tG
( )v t t
+
-( )C
eq tI v t
( ) ( ) ( )C Ct ti t t v t t v t
8
Equivalent Circuit Model-BE• Inductor
( ) ( ) ( )tLi t t i t v t t
+
L
-
+
-
( )v t t
( )i t t
( )i t t
Leq tR
( )v t t
+
-( )L
eq tV i t
( ) ( ) ( )L Lt tv t t i t t i t
9
Equivalent Circuit Model-TR• Capacitor
2( ) ( ) ( ( ) ( ))tCv t t v t i t i t t
+
C
-
+
-
( )v t t
( )i t t
( )i t t
2Ceq tG
( )v t t
+
- 2 ( ) ( )Ceq tI v t i t
2 2( ) ( ) ( ) ( )C Ct ti t t v t t v t i t
10
Equivalent Circuit Model-TR• Inductor
2( ) ( ) ( ( ) ( ))tLi t t i t v t v t t
+
L
-
+
-
( )v t t
( )i t t
( )i t t
2Leq tR
( )v t t
+
-2 ( ) ( )L
eq tV i t v t
2 2( ) ( ) ( ) ( )L Lt tv t t i t t i t v t
Trap Rule, Forward-Euler, Backward-Euler All are one-step methods xk+1 is computed using only xk, not xk-1, xk-2, xk-3... Forward-Euler is the simplest No equation solution explicit method. Backward-Euler is more expensive Equation solution each step implicit method most stable (FE/BE/TR) Trapezoidal Rule might be more accurate Equation solution each step implicit method More accurate but less stable, may cause oscillation
Summary of Basic Concepts
11
12
Stabilities
Froward Euler
0 -1
h
stable
unstable
j
1k k k
k k
x x hx
x x
1k k kx x h x
11 0(1 ) (1 )kk kx h x h x
Difference EqnStability region 1-1
1z h
Im(z)
Re(z)
Im
Re
Forward Euler
ODE stability region
2
t
Region ofAbsolute Stability
FE region of absolute stability
13
14
Stabilities
Backward Euler
1 1
1 1
k k k
k k
x x hx
x x
1 1k k kx x h x
11 0
1 1( )
1 1k
k kx x xh h
0 1
h
1-h
stable
unstable
j
Difference EqnStability region 1-1
Im(z)
Re(z)
Im Backward Euler 1
1z h
Region ofAbsolute Stability
BE region of absolute stability
15
16
Stabilities
Trapezoidal
1 1
1 1
( )2k k k k
k k
k k
hx x x x
x x
x x
1 1( )2k k k k
hx x x x
11 0
1 12 2( )
1 12 2
kk k
h h
x x xh h
0 1
h
1+h/2
stable
unstable
-1
1-h/2
j
Convergence• Consistency: A method of order p (p>1) is
consistent if
• Stability: A method is stable if:
• Convergence: A method is convergent if:
Consistency + Stability Convergence
17
A-Stable
• Dahlqnest Theorem:– An A-Stable LMS (Linear MultiStep) method
cannot exceed 2nd order accuracy
• The most accurate A-Stable method (smallest truncation error) is trapezoidal method.
18
Convergence Analysis: Truncation Error
• Local Truncation Error (LTE):– At time point tk+1 assume xk is exact, the difference between
the approximated solution xk+1 and exact solution x*k+1 is called
local truncation error.
– Indicates consistancy
– Used to estimate next time step size in SPICE
• Global Truncation Error (GTE):– At time point tk+1, assume only the initial condition x0 at time t0
is correct, the difference between the approximated solution xk+1 and the exact solution x*
k+1 is called global truncation error.
– Indicates stability
19
LTE Estimation: SPICE• Taylor Expansion of xn+1 about the time point tn:
• Taylor Expansion of dxn+1/dt about the time point tn:
• Eliminate term in above two equations we get the trapezoidal rule
LTE 20
Time Step Control: SPICE• We have derived the local truncation error
the unit is charge for capacitor and flux for inductor
• Similarly, we can derive the local truncation error in terms of (1)
the unit is current for capacitor and voltage for inductor
• Suppose ED represents the absolute value of error that is allowed per time point. That is
together with (1) we can calculate the time step as
21
Time Step Control: SPICE (cont’d)• DD3(tn+1) is called 3rd divided difference, which is given
by the recursive formula
22