1 / 23

# Numerical Integration - PowerPoint PPT Presentation

Numerical Integration. CSE245 Lecture Notes. Content. Introduction Linear Multistep Formulae Local Error and The Order of Integration Time Domain Solution of Linear Networks. Transient analysis is to obtain the transient response of the circuits.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about ' Numerical Integration' - lana-kane

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### Numerical Integration

CSE245 Lecture Notes

• Introduction

• Linear Multistep Formulae

• Local Error and The Order of Integration

• Time Domain Solution of Linear Networks

Equations for transient analysis are usually differential equations.

Numerical integration: calculate the approximate solutions Xn.

Linear multistep formulae are the primary numerical integration method.

Introduction

k the circuits.

k

iXn-i + h iXn-i = 0

i=0

i=0

Linear Multistep Formulae

• Differential equations are

X = F(X)

• Assume values Xn-1, Xn-2, … , Xn-k and derivatives Xn-1, Xn-2, … , Xn-k are known, the solution Xn and Xn can be approximated by a polynomial of these values:

Linear Multistep Formulae the circuits.

• There are two distinct classes LMS:

• Explicit predictors

--- 0 = 0

--- Xn is the only unknown variable

• Implicit

--- 0 0

--- Xn, Xn are all unknown variables.

Linear Multistep Formulae the circuits.

• Three simplest LMS formulae:

• The forward Euler

• The backward Euler

• Trapezoidal

X(t) the circuits.

Xn

X(tn)

Xn-1

tn

tn-1

t

Linear Multistep Formulae

• The forward Euler

Xn– Xn-1– h Xn-1 = 0

where 0 = 1, 1 = -1, 0 = 0, 1 = -1

Linear Multistep Formulae the circuits.

• The backward Euler

Xn– Xn-1– h Xn = 0

where 0 = 1, 1 = -1, 0 = -1, 1 = 0

• It is an implicit representation. We may assume some initial value for Xn and iterate to approximate the solution Xn and Xn.

Linear Multistep Formulae the circuits.

• Trapezoidal

Xn– Xn-1– h (Xn + Xn-1 )/2= 0

where 0 = 1, 1 = -1, 0 = -1/2, 1 = -1/2

• It is also an implicit representation. Xn, Xn can be obtained through some iterative procedure.

Local Error the circuits.

• Two crucial concepts

• Local error --- the error introduced in a single step of the integration routine.

• Global error--- the overall error caused by repeated application of the integration formula.

Diverging flow the circuits.

X(t)

Converging flow

t

Global error and local error

Local Error

Local Error the circuits.

• Two types of error in each step:

• Round-offerror --- due to the finite-precision (floating-point) arithmetic.

• Truncationerror --- caused by truncation of the infinite Taylor series, present even with infinite-precision arithmetic.

k the circuits.

k

iX(tn-i) + h iX(tn-i)

i=1

i=0

Local Error and Order of Integration

• Local error Ek for LMS

Ek = X(tn) +

• Ek can be expanded into Taylor series. If the coefficients of the first pth derivatives are zero, the order of integration is p.

k the circuits.

k

i((tn-tn-i)/h)l + h (-l/h)i((tn-tn-i)/h)l-1

i=0

i=0

k

i = 0

i=0

k

k

(ii - i) = 0

[(ii - pi)ip-1] = 0

i=0

i=0

Order of Integration

• Let X(t) = ((tn-t)/h)l and tn– tn-i = ih,

• Ek =

• For pth order integration, the first p+1 elements (l = 0, 1, … , p) will all be zeros:

• l = 0

• l = 1

• l = p

Order of Integration the circuits.

• The forward Euler

0 = 1, 1 = -1, 0 = 0, 1 = -1

So l = 0 0 + 1 = 1 + (-1) = 0;

l = 1 00 + 11 - 0 - 1 = 10 + (-1)1 - 0 – (-1) = 0;

l = 2 (11 - 21)1 = ((-1)1 - 2(-1))1 = 1  0;

The forward Euler is 1th order.

Order of Integration the circuits.

• The backward Euler

0 = 1, 1 = -1, 0 = -1, 1 = 0

So l = 0 0 + 1 = 1 + (-1) = 0;

l = 1 00 + 11 - 0 - 1 = 10 + (-1)1 - (-1) - 0 = 0;

l = 2 (11 - 21)1 = ((-1)1 - 20)1 = -1  0;

The backward Euler is 1th order.

Order of Integration the circuits.

• Trapezoidal

0 = 1, 1 = -1, 0 = -1/2, 1 = -1/2

So l = 0 0 + 1 = 1 + (-1) = 0;

l = 1 00 + 11 - 0 - 1 = 10 + (-1)1 - (-1/2) – (-1/2) = 0;

l = 2 (11 - 21)1 = ((-1)1 - 2(-1/2))1 = 0;

l = 3 (11 - 31)12 = ((-1)1 - 3(-1/2))1 = 1/2  0;

The trapezoidal method is 2th order

Order of Integration the circuits.

• The algorithm for defining  and  :

--- Choose p, the order of the numerical integration method needed;

--- Choose k, the number of previous values needed;

--- Write down the (p+1) equations of pth order accuracy;

--- Choose other (2k-p) constrains of the coefficients  and ;

--- Combine and solve above (2k+1) equations;

--- Get the result coefficients  and .

k the circuits.

k

iXn-i + h iXn-i = 0

i=0

i=0

Solution of Linear Networks

• Combine the differential equations for linear networks and the numerical integration equations:

MX = -GX + Pu

(1)

(2)

k the circuits.

k

iXn-i + h iXn-i = 0

i=1

i=1

Solution of Linear Networks

(1)  Xn + h0Xn +

 Xn + h0Xn + b = 0

 Xn = (-1/h0)( Xn + b)

(2)+(3)M[(-1/h0)( Xn + b)] = -GXn + Pu

 (-1/h0) Xn = -GXn + Pu + (M/h0)b

(3)

i the circuits.c

(-C/h0)

– (C/h0) bc

vc

ic

vc

Solution of Linear Networks

• For capacitance

C vc = ic

 C [(-1/h0)( vc + bc)] = ic

 (-C/h0) vc– (C/h0) bc = ic

i the circuits.l

– (L/h0) bl

+

-

vl

(-L/h0)

il

vl

Solution of Linear Networks

• For inductance

L il = vl

 L [(-1/h0)( il + bl)] = vl

 (-L/h0) il– (L/h0) bl = vl

References the circuits.

• CK. Cheng, John Lillis, Shen Lin and Norman Chang

“Interconnect Analysis and Synthesis”, Wiley and Sons, 2000

• Jiri Vlach and Kishore Singhal

“Computer Methods for Circuit Analysis and Design”, 1983