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ELU 4Yfu
rL
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a_x
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Ccrr<sop ,"r)L[^y larte Wc*er- veprc'Vtn l'''he'L'-
Y[.v) =L-ye$ -^Yt-r,lgt I
Y(3) =] y'" YGI
Y{il=5.: pt /D= exL())
E
f ^1 Y
I 'lrr-ud I
L tJ 'z-
Il, v-
f-r,t,.,l i-
; i'I
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l--rt*,J= kl --Ul-r,J
q-
" *tl-v, -l
U1,*) = [-.,'2 71 "'{d L *rL'"zJ
o=o
--,
1 t 2t L^^)
,/
sJe yu"'c'](https://image.slidesharecdn.com/explicitimpliciteulerintegrationinmatlab-180723163255/85/MATLAB-Output-Voltage-of-Series-RC-Circuit-utilizing-Euler-Approximations-1-320.jpg)
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Jtru)d4
Urn
'dart n d''a''r^-n' '$
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Y0)=
[t{7)
{-,: 7 G
i:', l=l
U,J lr
9k) =L /
U
( x '*d s, t )'Y(Y) = (8 9' u B) ,'i'rd): rar*d* - tir i,; ' r(Y)
fiu) -WuLt)- r!{ts -{DlJ(l r
'l*= )u,Lt) -e7@ - !1k)'ale
,
Y, =Y uY,
_Qot
=)u--ay -SYdo
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o
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x,
D
A
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+
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I
I
t
t
t
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i
ue7rn,',h3
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i
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t
I
I
J-tl
{'r i,.-r,,.
l/Vr,,.,,,,) ?,,,,} { {nrl) = X( n)+ dL 't-(v1}- ) .r
Trn'r) 6 + €) -'To[n) + E"r6tort)1
fu r6htof
X{nrr):}Ln) * fl*h) + ultr }-t
rn = roG) - +Lt"'i",,,-* tc'(,1) -LcGt') 'kL (r,*'t1th
fob,vD =*&.)( - ;1J+ ft"(,)-ftrocn+r
. *kk6nrr)
#) = r:{.nr(,- fo) * ikfll*- -;(^tt) + n G.'-)
=, !sl**,(-k) . &.(*tl^:*"%tn*)
+( n..'il
L tt-r.u-4) -
qr'{
tlrvt)
rfrcJ = 16
f r- ft) oroL*.,) + ,
-/Y=," (rt*-r*^5tnr) {-r3t,.*" ))
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r L =:_ t r-r_qqc/](https://image.slidesharecdn.com/explicitimpliciteulerintegrationinmatlab-180723163255/85/MATLAB-Output-Voltage-of-Series-RC-Circuit-utilizing-Euler-Approximations-3-320.jpg)
![F
8HW 3 KATRINA
''''"*,
clear all' close all' clc
3-l; C:1;
tau:(R*C); %e-tltau
T:O.2;
a}:t/(R*C) ; bO: l/ (R*C);
tfinal:10;
nf inal=round ( t f inaL lTl ;
alpha: (1- (a0*T) ) ;
beta= (b0*T) ;
vA(1):o;
for n:1:nfinal
t(n):(n-1)*T;
tn:t (n) ;
vA (n+1) =aIPha*vA(n) +beta;
v (n): (1-exP (-tn/tau) ) ;
end%n
t (nfinal+1 ) :nfinal*T;
tn:t (nfinal+1) ;
v (nf inat+1) : ( 1-exP 1-tP/tau)') ;
ptot (t, vA, ' ' r' )
trold on
pl-ot (t,v)-;t$l:i*:*i:li$;::!i;:_itilirm(itnrm) rm tvr ' , 'Fontsi ze' 'tL)
ti€*['fitr]ra'4;; tnnnt and e;ct So]utions"'FontSize"all
t'*ts,I-?,'i;-ifoatimtfitfrrnl ' itn ft*: o' i' 10' ' ' ' '
rso' ' 'Eontsi ze' '111
rert,{S, 1-55,'i;;i-tfittfttf ''-d-filq itt leq rm10
"'
Eontsize
"
11 )
teEttsrl'4r'itT rr 0'5 sec"'FontSize-' t11)'
text{5' a-25,'itvrm(ittrm) 1
-11-;;'-titt
/tau} )
"'
Fontsize" 11)
p].ot( t4-5L [1'7L''r'] -
;i;aiin-s-4-81' tL'ss 1'5s l"b')
BEES x'iqq€o* Etl..g
}EE 3 ETIRI}{A ],XTTI.E
c1c.t a1l, ffi at}, crc
Hl; Cd,
"rF(nrc)
, !te-t/6u
!{t -2;
EO{/tlic}, bo- :l./tRrc}t
Efi.Bl4o;
Bfltralsrd(tfial/T);
alBrE=t1-la0*u );
leta{DorT} t
vL{1)-0,
-: for F1:afiu1
!(r)-{r-1}'T;
m4lE);
g*tE+,.} q1pflB*vLta) +bcta'
;^trrL n = 0,5,10,'-',t50
- vt0,0<tJrO
I=O.5scr:
,14= 1r+ie)
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![%HW 3 KATRINA LITTLE
clear aII, close+a1I' clc
11:1; C:1 ;
tau:(R*C); %e-t/tau
T--o-2;
1pLl (R*C); bO: l/ (R*C) ;
tfina]-:10;
nfinal:round (tf inat/T) ;
al1fia:(1/ (1+a0*T) ) ;
Ueta:(b0*T) / (1+a0*T) ;
va(1)-4;
1sa p:lssfinal
t.(nl:(n-11rr;
lrts(nl;
rrn {nr1l alI[a*vA (n ) +beta ;
w(nl:(]'-eryt (-tn/tau) )'
endtn
t (nfinal+l ) <rfinal*t;
tn:t (nfinal+1) ;
, infin-f+l): (1-exP (-tnltau) ) ;
ptot (t, vA, ' ' r' )
hold cn
plot (t, v)
f.ml:i;:;.;;l;l.Gi:'i;::_itlkr)*ritnrm)
rm tvr',' Fontsize" 11)
litfe('afprox#te (ERR) and E*;t' solutions"' FontSize" 11)
[e8-(5'1-7,'itv-[EA]frntfitnirnl' itn rn: o'i'ro'"''rso"'Fontsize"LL)
text (5, 1-SS" fit'irn (ittrm)'
--d-iI;q
itt Ieq rm10
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Eontsi ze
"
Ll)
l
i
texE (5,1- ll' 'itT rn: 0 ' 5 sec' '
i"=t is, l--25, 'itvrm(j-ttrm)
plot(t4.51 ,[1.71,''t') -. -
;i;;iia.s-a-81, r1'ss L'5s L'b'
'EontSize' , 11)
ii-";f-ft11 /taul )','EontSize''11)
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t-
t-
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cH au, g!e* alr' &
&4r LE;
6E(!.c), *e-:/rar
H-Za
!!41(tr+C, , H= 1l talcl t
r'r..1dPi
diEal=M(tfdffl,
8IIF(lI {1d0*f,});
he=tbo{}l (1*q*t}t
Ya(1){,
:, for d:did
t (tr' - (i-rl 't;
E+(q),
Y*t4+1) altftr*{[)&ai
Y{[}-(1.4('t w}}:
%FRR](https://image.slidesharecdn.com/explicitimpliciteulerintegrationinmatlab-180723163255/85/MATLAB-Output-Voltage-of-Series-RC-Circuit-utilizing-Euler-Approximations-5-320.jpg)
![%HW 3 KATRINA LITTLE
clear all, close atl? clc
R=1; C:1;
tau:(R*C); %e-tltau
t:O.2;
aO:7/ (R*C) ; b0: 1/ (R*C);
tfinal:10;
nfinal:round (t finaL / T) t
"r;;;:
((1 - (ao*r/2))/(1+(ao*r/2))) ;
;;;": 2* 1(bo* ("t/2)) / (1+ bo* ('r/2));
vR(L):0;
for n:1:nfinal
t(n):(n-1) *T;
tn:t (n) ,
vA (n+1 ) :a-]-Pha*vA ( n ) +beta ;
v (n) : (I'-eq> (-tn/tau) ) ;
end*a
t (nf inal+1 ) :rlf iJral* T ;
tn4{nfina].+1);
oitiitt.r+11 : ( 1-exP { -tn/tau} ) ;
plot(trvAr''rr)
hold on
plot (t, v)
;**il:il::''lii;iii'Iil:iii:-ltlh' rm
t)ili)rmr rm rvr '
' ' Eon'Lsi ze' '.-a)
title ('appto"t*.t" (Trapezoidtli
';;;' Exact. Solutions' ' 'FontSi ze' 'LL)
text (5, a-7,'itv-{rmA}tr*titiirml ' itn rm:-o' i'io' " ''159r '
rrontSize" 11)
text (5, 1-5s,' iii;it*tittrm) '"d-ii;q
itt req rm10"' Fontsize"Llr
text(5, L-4,'i-tT rrn: 0'5 secr '
rFontsize'
'1l.1"
text (5, L.25,'itvrm(ittrm) : j1-;^t-itt /tau) ) '
"
Fontsize' '
11)
plot ( [4.5] , tl ' 71 ,' 't' )
;;;;iia-s-+-el, tr'ss 1'ss L'b' )
tmR
IEII 3 (TARINA LI?I:-E
clff all, !!4g; alr
B,=1; o=if;
ffi{*'C}, te-ty'tau
L
1
5
4-
5
6-
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iq-
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llr-
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*
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lzr -
1ar -
Jzr
1*r -
i2€ -
1,, -
1zff -
H.Z;
al!-llia'c); to-
Ef&.I40 t
afaaal'4ud(ltlM1/'fl;
*ta- ttt - (io't./z))/(1+(40'I/21))
F; 2'( {bo' {r/2) } / l[+ bo" (T/2] ] ) ;
r!,{al'{,
, for d{.:qfiqaf.
r{8, =(r-U *t;
ffi(t!t
va(aill tflttsryL([]#B;
;n)-tl...qrt-tq./t8l) '
sd*!
E tllfirat}!.)'4fiqtlrf r
s+ {EEidl.+i) ,
e{af,h{I-+l) = (l-eq! {-al6E} } ;
pbE {e,?4,
I . rr }
. YA(n),n=0,5,10""''t5l)
* v(tl0<t<t0
F= 0-S sEc
{o*(t+re}](https://image.slidesharecdn.com/explicitimpliciteulerintegrationinmatlab-180723163255/85/MATLAB-Output-Voltage-of-Series-RC-Circuit-utilizing-Euler-Approximations-6-320.jpg)
MATLAB: Output Voltage of Series RC Circuit utilizing Euler Approximations
- 1. ,9Ir *"S + L y t^f ho*"i nrr Ult(,, ril,,>O ELU 4Yfu rL Crrc,-y,wm FU a_x Gtrn, DC,n-*".-,.- ^ ocntro aN cr-l Q-w. trv*r-lat'th Ccrr<sop ,"r)L[^y larte Wc*er- veprc'Vtn l'''he'L'- Y[.v) =L-ye$ -^Yt-r,lgt I Y(3) =] y'" YGI Y{il=5.: pt /D= exL()) E f ^1 Y I 'lrr-ud I L tJ 'z- Il, v- f-r,t,.,l i- ; i'I ",1 + l--rt*,J= kl --Ul-r,J q- " *tl-v, -l U1,*) = [-.,'2 71 "'{d L *rL'"zJ o=o --, 1 t 2t L^^) ,/ sJe yu"'c'
- 2. cla.-d,a, '/-t-'a-. OC''f Jtru)d4 Urn 'dart n d''a''r^-n' '$ '"rsry!'b s5 '<rw%/'rd;r/ Y0)= [t{7) {-,: 7 G i:', l=l U,J lr 9k) =L / U ( x '*d s, t )'Y(Y) = (8 9' u B) ,'i'rd): rar*d* - tir i,; ' r(Y) fiu) -WuLt)- r!{ts -{DlJ(l r 'l*= )u,Lt) -e7@ - !1k)'ale , Y, =Y uY, _Qot =)u--ay -SYdo : Jt l" o =Y, x, D A ," '6 ;:[:1.8]uk) *) -1 c)L L k-L* --'. O r, J Yat , ) [_r,( u Uz) O t*/o) ytr) r Y2
- 3. Kc*;" &c tnz = I - rk"--t t6 ec. 4- ry;LO-TLt) + qD i'l,i I I t t t * i ue7rn,',h3 "*R; ='WT r'ffi ''.' , .rt i lyea; r , $. " ;i.d.n.^.. .n Jo-h.* 9*" a-12rc v rrr tNA'""y i +L-{ io r-,-.t- ,. f, L+' ) " "**?* I tnr^* a") ri* Cexprc't ,0''=Lr-' {'ft)-- t^) +-- u''-^) rl'{tn! y ,^.,n=.r-* )* r5cnrri ='r'oi"l r TEL 1[") 'p-c ']* 'J ' uhn) =Erd(i -E) + s r'cn) i - 'ov '/L I *rL J ?r t I I J-tl {'r i,.-r,,. l/Vr,,.,,,,) ?,,,,} { {nrl) = X( n)+ dL 't-(v1}- ) .r Trn'r) 6 + €) -'To[n) + E"r6tort)1 fu r6htof X{nrr):}Ln) * fl*h) + ultr }-t rn = roG) - +Lt"'i",,,-* tc'(,1) -LcGt') 'kL (r,*'t1th fob,vD =*&.)( - ;1J+ ft"(,)-ftrocn+r . *kk6nrr) #) = r:{.nr(,- fo) * ikfll*- -;(^tt) + n G.'-) =, !sl**,(-k) . &.(*tl^:*"%tn*) +( n..'il L tt-r.u-4) - qr'{ tlrvt) rfrcJ = 16 f r- ft) oroL*.,) + , -/Y=," (rt*-r*^5tnr) {-r3t,.*" )) / F I l >l r L =:_ t r-r_qqc/
- 4. F 8HW 3 KATRINA ''''"*, clear all' close all' clc 3-l; C:1; tau:(R*C); %e-tltau T:O.2; a}:t/(R*C) ; bO: l/ (R*C); tfinal:10; nf inal=round ( t f inaL lTl ; alpha: (1- (a0*T) ) ; beta= (b0*T) ; vA(1):o; for n:1:nfinal t(n):(n-1)*T; tn:t (n) ; vA (n+1) =aIPha*vA(n) +beta; v (n): (1-exP (-tn/tau) ) ; end%n t (nfinal+1 ) :nfinal*T; tn:t (nfinal+1) ; v (nf inat+1) : ( 1-exP 1-tP/tau)') ; ptot (t, vA, ' ' r' ) trold on pl-ot (t,v)-;t$l:i*:*i:li$;::!i;:_itilirm(itnrm) rm tvr ' , 'Fontsi ze' 'tL) ti€*['fitr]ra'4;; tnnnt and e;ct So]utions"'FontSize"all t'*ts,I-?,'i;-ifoatimtfitfrrnl ' itn ft*: o' i' 10' ' ' ' ' rso' ' 'Eontsi ze' '111 rert,{S, 1-55,'i;;i-tfittfttf ''-d-filq itt leq rm10 "' Eontsize " 11 ) teEttsrl'4r'itT rr 0'5 sec"'FontSize-' t11)' text{5' a-25,'itvrm(ittrm) 1 -11-;;'-titt /tau} ) "' Fontsize" 11) p].ot( t4-5L [1'7L''r'] - ;i;aiin-s-4-81' tL'ss 1'5s l"b') BEES x'iqq€o* Etl..g }EE 3 ETIRI}{A ],XTTI.E c1c.t a1l, ffi at}, crc Hl; Cd, "rF(nrc) , !te-t/6u !{t -2; EO{/tlic}, bo- :l./tRrc}t Efi.Bl4o; Bfltralsrd(tfial/T); alBrE=t1-la0*u ); leta{DorT} t vL{1)-0, -: for F1:afiu1 !(r)-{r-1}'T; m4lE); g*tE+,.} q1pflB*vLta) +bcta' ;^trrL n = 0,5,10,'-',t50 - vt0,0<tJrO I=O.5scr: ,14= 1r+ie) %FRR
- 5. %HW 3 KATRINA LITTLE clear aII, close+a1I' clc 11:1; C:1 ; tau:(R*C); %e-t/tau T--o-2; 1pLl (R*C); bO: l/ (R*C) ; tfina]-:10; nfinal:round (tf inat/T) ; al1fia:(1/ (1+a0*T) ) ; Ueta:(b0*T) / (1+a0*T) ; va(1)-4; 1sa p:lssfinal t.(nl:(n-11rr; lrts(nl; rrn {nr1l alI[a*vA (n ) +beta ; w(nl:(]'-eryt (-tn/tau) )' endtn t (nfinal+l ) <rfinal*t; tn:t (nfinal+1) ; , infin-f+l): (1-exP (-tnltau) ) ; ptot (t, vA, ' ' r' ) hold cn plot (t, v) f.ml:i;:;.;;l;l.Gi:'i;::_itlkr)*ritnrm) rm tvr',' Fontsize" 11) litfe('afprox#te (ERR) and E*;t' solutions"' FontSize" 11) [e8-(5'1-7,'itv-[EA]frntfitnirnl' itn rn: o'i'ro'"''rso"'Fontsize"LL) text (5, 1-SS" fit'irn (ittrm)' --d-iI;q itt Ieq rm10 "' Eontsi ze " Ll) l i texE (5,1- ll' 'itT rn: 0 ' 5 sec' ' i"=t is, l--25, 'itvrm(j-ttrm) plot(t4.51 ,[1.71,''t') -. - ;i;;iia.s-a-81, r1'ss L'5s L'b' 'EontSize' , 11) ii-";f-ft11 /taul )','EontSize''11) ) *i= E _E=3r' - -Ea* f,- t- t- [- u- 14- 15- 1€- LI ls^llc- l.-la- t2- lrt - *i tJltn trm +m 3 &rRrm !]ro cH au, g!e* alr' & &4r LE; 6E(!.c), *e-:/rar H-Za !!41(tr+C, , H= 1l talcl t r'r..1dPi diEal=M(tfdffl, 8IIF(lI {1d0*f,}); he=tbo{}l (1*q*t}t Ya(1){, :, for d:did t (tr' - (i-rl 't; E+(q), Y*t4+1) altftr*{[)&ai Y{[}-(1.4('t w}}: %FRR
- 6. %HW 3 KATRINA LITTLE clear all, close atl? clc R=1; C:1; tau:(R*C); %e-tltau t:O.2; aO:7/ (R*C) ; b0: 1/ (R*C); tfinal:10; nfinal:round (t finaL / T) t "r;;;: ((1 - (ao*r/2))/(1+(ao*r/2))) ; ;;;": 2* 1(bo* ("t/2)) / (1+ bo* ('r/2)); vR(L):0; for n:1:nfinal t(n):(n-1) *T; tn:t (n) , vA (n+1 ) :a-]-Pha*vA ( n ) +beta ; v (n) : (I'-eq> (-tn/tau) ) ; end*a t (nf inal+1 ) :rlf iJral* T ; tn4{nfina].+1); oitiitt.r+11 : ( 1-exP { -tn/tau} ) ; plot(trvAr''rr) hold on plot (t, v) ;**il:il::''lii;iii'Iil:iii:-ltlh' rm t)ili)rmr rm rvr ' ' ' Eon'Lsi ze' '.-a) title ('appto"t*.t" (Trapezoidtli ';;;' Exact. Solutions' ' 'FontSi ze' 'LL) text (5, a-7,'itv-{rmA}tr*titiirml ' itn rm:-o' i'io' " ''159r ' rrontSize" 11) text (5, 1-5s,' iii;it*tittrm) '"d-ii;q itt req rm10"' Fontsize"Llr text(5, L-4,'i-tT rrn: 0'5 secr ' rFontsize' '1l.1" text (5, L.25,'itvrm(ittrm) : j1-;^t-itt /tau) ) ' " Fontsize' ' 11) plot ( [4.5] , tl ' 71 ,' 't' ) ;;;;iia-s-+-el, tr'ss 1'ss L'b' ) tmR IEII 3 (TARINA LI?I:-E clff all, !!4g; alr B,=1; o=if; ffi{*'C}, te-ty'tau L 1 5 4- 5 6- 1t- iq- lro llr- tfl 113 - Itr - ltr - !re * lrr 1u- 1."- l?0 * 1zr - lzr - 1ar - Jzr 1*r - i2€ - 1,, - 1zff - H.Z; al!-llia'c); to- Ef&.I40 t afaaal'4ud(ltlM1/'fl; *ta- ttt - (io't./z))/(1+(40'I/21)) F; 2'( {bo' {r/2) } / l[+ bo" (T/2] ] ) ; r!,{al'{, , for d{.:qfiqaf. r{8, =(r-U *t; ffi(t!t va(aill tflttsryL([]#B; ;n)-tl...qrt-tq./t8l) ' sd*! E tllfirat}!.)'4fiqtlrf r s+ {EEidl.+i) , e{af,h{I-+l) = (l-eq! {-al6E} } ; pbE {e,?4, I . rr } . YA(n),n=0,5,10""''t5l) * v(tl0<t<t0 F= 0-S sEc {o*(t+re}