Numerical Integration

CSE245 Lecture Notes

Content

Introduction Linear Multistep Formulae Local Error and The Order of

Integration Time Domain Solution of Linear

Networks

Introduction Transient analysis is to obtain the

transient response of the circuits. Equations for transient analysis are

usually differential equations. Numerical integration: calculate the

approximate solutions Xn. Linear multistep formulae are the

primary numerical integration method.

Linear Multistep Formulae Differential equations are

X = F(X) Assume values Xn-1, Xn-2, … , Xn-k and

derivatives Xn-1, Xn-2, … , Xn-k are known, the solution Xn and Xn can be approximated by a polynomial of these values: iXn-i + h iXn-i = 0

i=0

k

i=0

k

Linear Multistep Formulae

There are two distinct classes LMS: Explicit predictors

--- 0 = 0

--- Xn is the only unknown variable Implicit

--- 0 0

--- Xn, Xn are all unknown variables.

Linear Multistep Formulae

Three simplest LMS formulae: The forward Euler The backward Euler Trapezoidal

Linear Multistep Formulae The forward Euler

Xn – Xn-1 – h Xn-1 = 0

where 0 = 1, 1 = -1, 0 = 0, 1 = -1

tn-1 tn

Xn-1

Xn X(tn)

X(t)

t

Linear Multistep Formulae

The backward EulerXn – Xn-1 – h Xn = 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 Xn.

Linear Multistep Formulae

TrapezoidalXn – Xn-1 – h (Xn + Xn-1 )/2= 0

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

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

Local Error

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.

Local Error

X(t)

t

Global error and local error

Converging flow

Diverging flow

Local Error

Two types of error in each step: Round-off error --- due to the finite-

precision (floating-point) arithmetic. Truncation error --- caused by

truncation of the infinite Taylor series, present even with infinite-precision arithmetic.

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.

iX(tn-i) + h iX(tn-i) i=1

k

i=0

k

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

i((tn-tn-i)/h)l + h (-l/h) i ((tn-tn-i)/h)l-1 i=0

k

i=0

k

i = 0i=0

k

(ii - i) = 0i=0

k

[(ii - pi)ip-1] = 0i=0

k

Order of Integration

The forward Euler0 = 1, 1 = -1, 0 = 0, 1 = -1 So l = 0 0 + 1 = 1 + (-1) = 0;

l = 1 00 + 11 - 0 - 1 = 10 + (-1)1 - 0 – (-1) = 0;

l = 2 (11 - 21)1 = ((-1)1 - 2(-1))1 = 1 0;

The forward Euler is 1th order.

Order of Integration

The backward Euler0 = 1, 1 = -1, 0 = -1, 1 = 0

So l = 0 0 + 1 = 1 + (-1) = 0;l = 1 00 + 11 - 0 - 1 = 10 + (-1)1 -

(-1) - 0 = 0;l = 2 (11 - 21)1 = ((-1)1 - 20)1 = -1

0;

The backward Euler is 1th order.

Order of Integration

Trapezoidal0 = 1, 1 = -1, 0 = -1/2, 1 = -1/2So l = 0 0 + 1 = 1 + (-1) = 0;

l = 1 00 + 11 - 0 - 1 = 10 + (-1)1 - (-1/2) – (-1/2) = 0;

l = 2 (11 - 21)1 = ((-1)1 - 2(-1/2))1 = 0;l = 3 (11 - 31)12 = ((-1)1 - 3(-1/2))1 = 1/2 0;

The trapezoidal method is 2th order

Order of Integration 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 .

Solution of Linear Networks

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

MX = -GX + Pu

iXn-i + h iXn-i = 0 i=0

k

i=0

k(1)

(2)

Solution of Linear Networks

(1) Xn + h0Xn +

Xn + h0Xn + b = 0

Xn = (-1/h0)( Xn + b)

(2)+(3) M[(-1/h0)( Xn + b)] = -GXn + Pu

(-1/h0) Xn = -GXn + Pu +

(M/h0)b

iXn-i + h iXn-i = 0 i=1

k

i=1

k

(3)

Solution of Linear Networks For capacitance

C vc = ic

C [(-1/h0)( vc + bc)] = ic

(-C/h0) vc – (C/h0) bc = ic

ic

vc

vc

ic

– (C/h0) bc (-C/h0)

Solution of Linear Networks For inductance

L il = vl

L [(-1/h0)( il + bl)] = vl

(-L/h0) il – (L/h0) bl = vl

il

vl

+-

il

– (L/h0) bl

(-L/h0)

vl

References 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