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Numerical Integration CSE245 Lecture Notes
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Content Introduction Linear Multistep Formulae Local Error and The Order of Integration Time Domain Solution of Linear Networks
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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.
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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
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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.
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Linear Multistep Formulae Three simplest LMS formulae: The forward Euler The backward Euler Trapezoidal
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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
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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.
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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.
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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.
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Local Error X(t) t Global error and local error Converging flow Diverging flow
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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.
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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
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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
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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.
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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.
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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
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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 .
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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)
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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)
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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 )
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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
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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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