Time-domain Analysis of Linear and Nonlinear Circuits
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1 Tme-doman Analyss of Lnear and Nonlnear Crcus Dr. José Erneso Rayas-Sáncez February 4, 8 Tme-doman Analyss of Lnear and Nonlnear Crcus Dr. José Erneso Rayas-Sáncez Inroducon Tme doman analyss can be realzed n e ransen regme or n e seady-sae regme Calculang e ransen response of a crcu mples solvng a sysem of dfferenal equaons A number of meods can be used o calculae e me response of elecrcal crcus: Lnear Mul-Sep LMS formulae Runge-Kua R-K formulae armonc-balance B Dr. J.E. Rayas-Sáncez
2 Tme-doman Analyss of Lnear and Nonlnear Crcus Dr. José Erneso Rayas-Sáncez February 4, 8 Solvng Dfferenal Equaons d f, d f, + + Forard Euler formula Dr. J.E. Rayas-Sáncez 3 Solvng Dfferenal Equaons con. + + Backard Euler formula Dr. J.E. Rayas-Sáncez 4
3 Tme-doman Analyss of Lnear and Nonlnear Crcus Dr. José Erneso Rayas-Sáncez February 4, 8 Solvng Dfferenal Equaons con. + + Trapezodal formula Dr. J.E. Rayas-Sáncez 5 Generalzng o Sysems of Dfferenal Eq. f, f,,, n f Forard Euler: Backard Euler: Trapezodal: Dr. J.E. Rayas-Sáncez 6 3
4 Tme-doman Analyss of Lnear and Nonlnear Crcus Dr. José Erneso Rayas-Sáncez February 4, 8 Usng Predcors and Correcors Backard Euler and Trapezodal formulae requre + o calculae +, bu s value s no knon a e eraon Forard Euler can be used as a predcor for +, c can be laer nsered no a correcor usng Backard Euler or Trapezodal formulae Dr. J.E. Rayas-Sáncez 7 A Forard Euler Malab Implemenaon % Ts funcon solves a sysem of n frs-order ordnary dfferenal % equaons =f, usng e Forard Euler formula. % % Usage: [,] = F_Eulerfun,,,f, % : Independen varable ro vecor of leng N. % : Soluon of e dfferenal equaons an N by n mar. % fun: name of e muldmensonal vecor funcon srng a % evaluaes e dfferenal equaons. Ts funcon akes o % argumens: a ro vecor of leng n and a scalar ndependen % varable. I reurns a ro vecor f of leng n. % : nal condon ro vecor, of leng n. = =. % : nal value of e ndependen varable. % f: fnal value of e ndependen varable. % : sep used for e ndependen varable. funcon [,] = F_Eulerfun,,,f, N = roundf-/; % Number of pons. n = leng; = zerosn+,n; = zeros,n+;,: = ; = ; = ; le <=N f = fevalfun,,:,; +,: =,: + *f; + = + ; = +; end Dr. J.E. Rayas-Sáncez 8 4
5 Tme-doman Analyss of Lnear and Nonlnear Crcus Dr. José Erneso Rayas-Sáncez February 4, 8 A Backard Euler Malab Implemenaon % Ts funcon solves a sysem of n frs-order ordnary dfferenal equaons % =f, usng e Backard Euler formula predcors bu no correcors. % % Usage: [,] = B_Eulerfun,,,f, % : Independen varable ro vecor of leng N+. % : Soluon of e dfferenal equaons an N+ by n mar. % fun: name of e muldmensonal vecor funcon srng a % evaluaes e dfferenal equaons. Ts funcon akes o % argumens: a ro vecor of leng n and a scalar ndependen % varable. I reurns a ro vecor f of leng n. % : nal condon ro vecor, of leng n. = =. % : nal value of e ndependen varable. % f: fnal value of e ndependen varable. % : sep used for e ndependen varable. funcon [,] = B_Eulerfun,,,f, N = roundf-/; % Number of pons. n = leng; = zerosn+,n; = zeros,n+;,: = ; = ; = ; le <=N f = fevalfun,,:,; p =,: + *f; + = + ; f = fevalfun,p,+; +,: =,: + *f; = +; end Dr. J.E. Rayas-Sáncez 9 A Trapezodal Malab Implemenaon % Ts funcon solves a sysem of n frs-order ordnary dfferenal equaons % =f, usng e Trapezodal formula predcors bu no correcors. % % Usage: [,] = Trapezodalfun,,,f, % : Independen varable ro vecor of leng N+. % : Soluon of e dfferenal equaons an N+ by n mar. % fun: name of e muldmensonal vecor funcon srng a % evaluaes e dfferenal equaons. Ts funcon akes o % argumens: a ro vecor of leng n and a scalar ndependen % varable. I reurns a ro vecor f of leng n. % : nal condon ro vecor, of leng n. = =. % : nal value of e ndependen varable. % f: fnal value of e ndependen varable. % : sep used for e ndependen varable. funcon [,] = Trapezodalfun,,,f, N = roundf-/; % Number of pons. n = leng; = zerosn+,n; = zeros,n+;,: = ; = ; = ; le <=N f = fevalfun,,:,; p =,: + *f; + = + ; fn = fevalfun,p,+; +,: =,: + /*f+fn; = +; end Dr. J.E. Rayas-Sáncez 5
6 Tme-doman Analyss of Lnear and Nonlnear Crcus Dr. José Erneso Rayas-Sáncez February 4, 8 Eample - Forard Euler = Forard Euler Eac Soluon Eac soluon: 3e Dr. J.E. Rayas-Sáncez Eample - Backard Euler = Backard Euler Eac Soluon Eac soluon: 3e Dr. J.E. Rayas-Sáncez 6
7 Tme-doman Analyss of Lnear and Nonlnear Crcus Dr. José Erneso Rayas-Sáncez February 4, 8 Eample - Trapezodal = Trapezodal Eac Soluon Eac soluon: 3e Dr. J.E. Rayas-Sáncez 3 Eample - Comparson =.5 Eac soluon: 3e errors F-Euler B-Euler Trapezodal Dr. J.E. Rayas-Sáncez 4 7
8 Tme-doman Analyss of Lnear and Nonlnear Crcus Dr. José Erneso Rayas-Sáncez February 4, 8 Eample - Comparson Decreasng =. 5 F-Euler B-Euler Trapezodal Eac soluon: 3e errors Dr. J.E. Rayas-Sáncez 5 Eample - Forard Euler 4 =. Eac soluon: e Forard Euler Eac Soluon Dr. J.E. Rayas-Sáncez 6 8
9 Tme-doman Analyss of Lnear and Nonlnear Crcus Dr. José Erneso Rayas-Sáncez February 4, 8 Eample - Backard Euler 4 =. Eac soluon: e Backard Euler Eac Soluon Dr. J.E. Rayas-Sáncez 7 Eample - Trapezodal 4 =. Eac soluon: e Trapezodal Eac Soluon Dr. J.E. Rayas-Sáncez 8 9
10 Tme-doman Analyss of Lnear and Nonlnear Crcus Dr. José Erneso Rayas-Sáncez February 4, 8 Eample - Comparson 4 =. Eac soluon: e 4 errors F-Euler B-Euler Trapezodal Dr. J.E. Rayas-Sáncez 9 Eample - Comparson Decreasng 4 =...5 Eac soluon: e 4 errors -.5 F-Euler B-Euler Trapezodal Dr. J.E. Rayas-Sáncez
11 Tme-doman Analyss of Lnear and Nonlnear Crcus Dr. José Erneso Rayas-Sáncez February 4, 8 Dr. J.E. Rayas-Sáncez Sably of Inegraon Tes dfferenal equaon: soluon: Forard Euler: Backard Euler: Trapezodal: e / / / / Dr. J.E. Rayas-Sáncez Lnear Sysems of Dfferenal Equaons Backard Euler: Trapezodal: B, f, f B B B B B B
12 Tme-doman Analyss of Lnear and Nonlnear Crcus Dr. José Erneso Rayas-Sáncez February 4, 8 Transen Soluon of Lnear Crcus Le e sysem equaons n e Laplace doman be s X W Takng e nverse Laplace Transform, Dr. J.E. Rayas-Sáncez 3 Transen Soluon of Lnear Crcus con. Forard Euler: mg be sngular Dr. J.E. Rayas-Sáncez 4
13 Tme-doman Analyss of Lnear and Nonlnear Crcus Dr. José Erneso Rayas-Sáncez February 4, Dr. J.E. Rayas-Sáncez Transen Soluon of Lnear Crcus con. Backard Euler: 6 Dr. J.E. Rayas-Sáncez Transen Soluon of Lnear Crcus con. Trapezodal:
14 Tme-doman Analyss of Lnear and Nonlnear Crcus Dr. José Erneso Rayas-Sáncez February 4, 8 MNA Formulaon for Transen Analyss Te MNA equaon can be formulaed ou usng orened graps or ncdence marces A and A and W can be drecly formulaed by nspecon, usng samps Once s knon, e L and C elemens can be separaed so a c can be solved n e me doman usng e prevous meods Dr. J.E. Rayas-Sáncez T AY A T Y A A V Z I n A J W s X W X W 7 4
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