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Numerical Integration CSE245 Lecture Notes. Content Introduction Linear Multistep Formulae Local Error and The Order of Integration Time Domain Solution.

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Presentation on theme: "Numerical Integration CSE245 Lecture Notes. Content Introduction Linear Multistep Formulae Local Error and The Order of Integration Time Domain Solution."— Presentation transcript:

1 Numerical Integration CSE245 Lecture Notes

2 Content Introduction Linear Multistep Formulae Local Error and The Order of Integration Time Domain Solution of Linear Networks

3 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 X n. Linear multistep formulae are the primary numerical integration method.

4 Linear Multistep Formulae Differential equations are X = F(X) Assume values X n-1, X n-2, …, X n-k and derivatives X n-1, X n-2, …, X n-k are known, the solution X n and X n can be approximated by a polynomial of these values:   i X n-i + h   i X n-i = 0 i=0 k k

5 Linear Multistep Formulae There are two distinct classes LMS:  Explicit predictors ---  0 = 0 --- Xn is the only unknown variable  Implicit ---  0  0 --- X n, X n are all unknown variables.

6 Linear Multistep Formulae Three simplest LMS formulae:  The forward Euler  The backward Euler  Trapezoidal

7 Linear Multistep Formulae The forward Euler X n – X n-1 – h X n-1 = 0 where  0 = 1,  1 = -1,  0 = 0,  1 = -1 t n-1 tntn X n-1 XnXn X(t n ) X(t) t

8 Linear Multistep Formulae The backward Euler X n – X n-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 X n and iterate to approximate the solution X n and X n.

9 Linear Multistep Formulae Trapezoidal X n – X n-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. X n, X n can be obtained through some iterative procedure.

10 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.

11 Local Error X(t) t Global error and local error Converging flow Diverging flow

12 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.

13 Local Error and Order of Integration Local error E k for LMS E k = X(t n ) + E k can be expanded into Taylor series. If the coefficients of the first pth derivatives are zero, the order of integration is p.   i X(t n-i ) + h   i X(t n-i ) i=1 k i=0 k

14 Order of Integration Let X(t) = ((t n -t)/h) l and t n – t n-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 ((t n -t n-i )/h) l + h (-l/h)   i ((tn-tn-i)/h) l-1 i=0 k k   i = 0 i=0 k  (  i i -  i ) = 0 i=0 k  [(  i i - p  i )i p-1 ] = 0 i=0 k

15 Order of Integration The forward Euler  0 = 1,  1 = -1,  0 = 0,  1 = -1 Sol = 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.

16 Order of Integration The backward Euler  0 = 1,  1 = -1,  0 = -1,  1 = 0 Sol = 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.

17 Order of Integration Trapezoidal  0 = 1,  1 = -1,  0 = -1/2,  1 = -1/2 Sol = 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

18 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 .

19 Solution of Linear Networks Combine the differential equations for linear networks and the numerical integration equations: MX = -GX + Pu   i X n-i + h   i X n-i = 0 i=0 k k (1) (2)

20 Solution of Linear Networks (1)  X n + h  0 X n +  X n + h  0 X n + b = 0  X n = (-1/h  0 )( X n + b) (2)+(3)  M[(-1/h  0 )( X n + b)] = -GX n + Pu  (-1/h  0 ) X n = -GX n + Pu + (M/h  0 )b   i X n-i + h   i X n-i = 0 i=1 k k (3)

21 Solution of Linear Networks For capacitance C v c = i c  C [(-1/h  0 )( v c + b c )] = i c  (-C/h  0 ) v c – (C/h  0 ) b c = i c icic vcvc vcvc icic – (C/h  0 ) b c (-C/h  0 ) 

22 Solution of Linear Networks For inductance L i l = v l  L [(-1/h  0 )( i l + b l )] = v l  (-L/h  0 ) i l – (L/h  0 ) b l = v l ilil vlvl + - ilil – (L/h  0 ) b l (-L/h  0 ) vlvl 

23 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


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