modelamiento de sistemas eléctricos y mecanicos

8/19/2019 Modelamiento de sistemas eléctricos y mecanicos

1/23

UNIVERSIDAD TECNICA DE ORUROFACULTAD NACIONAL DE INGENIERÍA

INGENIERÍA ELÉCTRICA – ELECTRÓNICA

Materia: ELT 2590 SISTEMAS DE CONTROL IA!i"iar: E#i"$ E"e%a Ri&era T'&arD'(e%te: I%)* Ra#ir' Fra%+ A"ie%,re Gar(-a

RESOLUCIÓN PRÁCTICA 2II/2015

. E%(e%tre "a/ %(i'%e/ ,e tra%/ere%(ia1 G ( s )=V o ( s )

V i, ara (a,a re, 3e /e

#e/tra e% "a 4)ra* M6t',' ,e %','/1 #6t',' ,e #a""a/7

SOLUCIÓN:

I%(i/' a7

8'r %','/

N',' a:

i2

i.

i

a


2/23

i1=i

2+i

3

i L1=i L2+i R

1 L

1∫ ( vi−v R ) dt = 1 L

2∫ ( v R−vC ) dt + 1 R v R m /m(derivando)

1

L1

( vi−v R )= 1

L2

( v R−vC )+ 1

R v́ R

1

L1

vi− 1

L1

v R= 1

L2

v R− 1

L2

vC + 1

R v́ R

1

L1

v i= 1

L2

v R+ 1

L1

v R+ 1

R v́ R−

1

L2

vC m/ m(aplicando T . L)

1

L1

V i =( 1 L2

+ 1

L1

+ 1

R s)V R− 1 L

2

V C

1

2V

i=( 13 + 12+ 11 s )V R−13V C

1

2V

i=( 13 + 12+ 11 s )V R−13V C

1

2V i=( 56 +s)V R−13 V C

V i=(53 +2 s )V R−23 V C (1)

Ta#i6% /ae#'/ 3e "a ('rrie%te e% %a #i/#a ra#a e/ i)a"1 e%t'%(e/:

i L2=iC


3/23

1

L2

∫ ( v R−vC ) dt =C ( v́C ) m/m(derivando )

1

L2

( v R−vC )=C ´(vC )m/m(aplicando T . L)

1

L2

V R=(C s2+ 1 L2 )V C

1

3V

R=(0.5s2+ 13 )V C

V R

=

(3

2s2+1

)V

C

(2)

A,e#;/ ,e" )r;4('1 te%e#'/ 3e:

vo= 1

L2

∫ ( v R−vC ) dt

v́o= 1

L2

( v R −vC )

s V o= 1

L2

V R− 1

L2

V C

s V o=1

3V R−

1

3V C (3)

Ree#"a+a%,' 27 e% .7

V i=(5

3 +2 s )(3

2 s

2

+1)V C −

2

3 V C

V i=(3 s3+ 52 s2+2 s+1)V C (4)

Ree#"a+a%,' 27 e% 7


4/23

s V o=1

3 (32 s2+1)V C −13 V C

s V o=

(

1

2s2

)V C

V o=1

2s V C

2

s V o=V C (5)

Ree#"a+a%,' 57 e%


5/23

8'r #a""a/

Ma""a .

v1=2 í1+1 (i1−i2 )

V 1=(2 s +1 ) I

1− I

2(1 )

Ma""a 2

0=3 í2+2∫ i2 dt +1(i2−i1 )

0=3 í2+2 i

2+1 (í2−í1 )

0=(3 s2+s+2 ) I 2−s I 1

I 1=(3 s2+s+2 ) I 2

s (2)

De" r'"e#a1 te%e#'/:

vo=3 í2

V o=3 s I 2 (3 )

I 2=V o

3 s

Ree#"a+a%,' 2 7 e% .7:

V 1=(2 s +1 )

(3 s2+s +2 ) I 2s

− I 2

i2i.


6/23

V 1=( (2 s+1 ) (3s

2+s+2 )s

−1) I 2

V 1=

(6 s

3+5 s2+4 s +2

s

)V o

3 s

G ( s )=V o

V 1= 3

s2

6 s3+5 s2+4 s+2

2 Ote%)a "a %(i=% ,e tra%/ere%(ia Eo(s)/ Ei(s) ,e" (ir(it' e"6(tri(' 3e

/e #e/tra* M6t','/ ,e "'/ %','/ $ #a""a/7

M6t','/ ,e "'/ %','/:

N',' a

i1=i

3+i

2

ei−ea R1

=ea−eo

R2+1

L∫ eadt

R2 L éi− R2 L éa= R1 L éa− R1 L éo+ R1 R2 ea

R2 Ls E1=(( R2+ R1) Ls+ R1 R2 ) Ea− R1 Ls Eo(1)

Le)':

i R2=iC =i3

i

i2

i.


7/23

ea−eo R2

=C éo

ea−eo ¿ R2C éo

Ea= Eo ( R2Cs+1 )(2)

Ree#"a+a%,' 27 e% .7:

R2 Ls E1=(( R2+ R1) Ls+ R1 R2 ) E o ( R2 Cs+1)− R1 Ls E o

R2 Ls E

1=[( ( R2+ R1 ) Ls + R1 R2) ( R2Cs+1 )− R1 Ls ] Eo

Eo

E1=

R2 Ls

( ( R2+ R1 ) Ls+ R1 R2 ) ( R2Cs+1 )− R1 Ls

Eo

E1=

R2 Ls

R2

CL ( R1+ R2 ) s2+( L R2+ R1 R2

2C )s + R1 R2

R1+ R

LC (¿ ¿2)s2+(C R1 R

2+ L)s + R

1

Eo

E i=

Ls¿

M6t',' ,e #a""a/

Ma""a .


8/23

ei= R1i1+ L (í1−í2)

Ei=( Ls+ R1 ) I 1− Ls I 2(1)

Ma""a 2

0= L (í2−í1 )+ R2 i2+ 1

C ∫ i2 dt

0= Lí2− L í1+ R2 í2+ 1

C i2

0= I 2( L s2+ R2 s+ 1C )− L s2 I 1

I 1=CLs

2+CR2 s+1

CL s2

I 2(2)

Ree#"a+a%,' 27 e% .7:

Ei=[ ( Ls+ R1 ) CL s2+CR

2s+1

CLs2

− Ls] I 2(3)

D'%,e:

eo= 1

C ∫i2 dt

Cs Eo= I 2

Ree#"a+a%,' e% 7:

Ei=[ ( Ls+ R1 )CL s2+CR2 s+1

CLs2

− Ls]Cs Eo Eo

E i=

Ls

C L2

s3+ LC R

2s2+ Ls+ R

1CL s

2+C R1 R

2s+ R

1−C L2 s3


9/23

R1+ R

LC (¿ ¿2)s2+(C R1 R

2+ L)s + R

1

Eo

E i=

Ls¿

E% "a 4)ra /e #e/tra % (ir(it' e"6(tri('* Ot6%)a/e % /i/te#a ,ee(a(i'%e/ -%te)r'>,iere%(ia"e/ /i#"t;%ea/ 3e rere/e%te% a" (ir(it'*M6t',' ,e %','/1 #6t',' #a""a/7

M6t',' %','/:

N',' a:

v−va R1

=C 1

d

dt (va−vb )(1)

N',' :

i1=i

2+i

3

C 1

d

dt (va−vb )=

vb−vd R3

+ 1

L1∫ vb−vc dt (2)

N',' (:

1 L1∫ vb−vc dt = v

c

R2(3 )

N',' ,:

vb−vd R3

=C 2

d

dt ( vd )(4)

,

(

a

i2

i

i.


10/23

M6t','/ ,e #a""a/:

Ma""a .:

v (t )= R1i1+

1

C ∫ i1 dt + L1

d

dt (i1−i2 )+ R2 (i1−i2) (1

)

Ma""a 2:

0= L1d

dt (i2−i1 )+ R2 (i2−i1 )+ R3 (i2−i1 )+

1

C 2

∫ i2 dt (2)

< %a re, ,e e%te e% T /e ti"i+a re(e%te#e%te e% /i/te#a/ ,e ('%tr'" ,e(a ('#' %a re, ,e 4"tr'* E% "a 4)ra /e #e/tra e" (ir(it' ,e %a re, ,ee%te e% T* De#6/tre/e 3e "a %(i=% ,e tra%/ere%(ia ,e "a re, e/

V o(s)V ¿(s)

= 1+2 R1 Cs+ R1 R2C

2s2

1+(2 R1+ R2 )Cs+ R1 R2 C 2

s2

Di?@e/e e" ,ia)ra#a ,e '"'/>(er'/ (a%,' R1=0.5, R2=1 $ C =0.5 *

M6t',' ,e %','/:


11/23

N',' a:

C ( v́¿− v́a )= 1

R1

( va )+C ( v́a− v́o )

R1C v́¿= R1C v́a+ R1C v́a+va− R1C v́o

R1

Cs V ¿=(2 R1Cs+1 )V a− R1Cs V o(1)

N',' :

C ( v́a− v́o )= 1

R1

(vo−v¿)

R1Cs V a=(Cs+1 )V o−V ¿

V a=

(Cs+1 )V o−V ¿ R

1Cs

(2)

Ree#"a+a%,' 27 e% .7:

R1

Cs V ¿=(2 R1Cs+1 ) (Cs+1) V o−V ¿

R1Cs − R

1Cs V o

De/e@a%,' $ /i#"i4(a%,' te%e#'/:

V o(s)V ¿(s)

= 1+2 R1 Cs+ R1 R2C

2s2

1+(2 R1+ R2 )Cs+ R1 R2 C 2

s2

Ree#"a+a%,' ,at'/:

V o(s)V

¿

(s)=

1+0.5s+0.125 s2

1+s+0.125 s2 =

( s+2−2 j ) (s+2+2 j )(s+6.8)(s+1.2)


12/23

Maa ,e '"'/ $ (er'/ ,e" /i/te#a:

5 Ote%)a "a %(i=% ,e tra%/ere%(ia X 1(s)/ U (s) $ X 2(s)/ U (s) ,e"

/i/te#a #e(;%i(' 3e /e #e/tra e% "a 4)ra*

S'"(i=%:

u−fk 1−fb1−fk 3=m1 ´

1

u−k 1 1−b1 ´

1−k 3 ( 1− 2 )=m1 ´ 1

>.*2>*B

>@2

@2

Re

I#

FF.

F.


13/23

U =(m1 s2+b1 s +k 1+k 3) X 1−k 3 X 2(1)

fk 3−fk 2−fb2=m2 ´

2

k 3 ( 1− 2)−k 2 2−b2 ´ 2=m2 ´ 2

k 3 X 1=(m2 s

2+b2 s +k 2+k 3) X 2(2)

De/e@a%,' 2 ,e 27:

X 2=

k 3

(m2 s2+b2s+k 2+k 3)

X 1

Ree#"a+a%,' e% .7:

U =(m1 s2+b1 s +k 1+k 3) X 1−k 3 k 3(m2 s

2+b2 s +k 2+k 3) X 1

U =[ (m1 s2+b1 s+k 1+k 3)− k 32

(m2 s2+b2 s+k 2+k 3 ) ] X 1

X 1

U =

(m2 s2+b2 s+k 2+k 3)

(m1 s2+b1 s+k 1+k 3) (m2 s

2+b2 s+k 2+k 3)−k 32

Ree#"a+a%,' 27 e% .7:

U =(m1 s2+b1s+k 1+k 3) (m2 s

2+b2 s+k 2+k 3 ) X 2k 3

−k 3 X 2

F2

F2F


14/23

U =[ (m1 s2+b1 s+k 1+k 3) (m2 s2+ b2 s+ k 2+k 3)

k 3−k 3 ] X 2

X 2

U =

k 3

(m1 s2+b1 s+k 1+k 3) (m2 s

2+b2 s+k 2+k 3 )−k 32

8ara e" /i/te#a ,e "a 4)ra1 e%(e%tre "a %(i=% ,e tra%/ere%(ia

G ( s )= X 1(s)/ ! (s)

S'"(i=%:

f −fv 3−fv2−fk 2=m2 ´

2

f −v 3 ´ 2−v 2 ( ´ 2− ´ 1 )−k 2( 2− 1 )=m2 ´ 2

! =(m2 s2+(v 3+v2)s+k 2) X 2−(v 2 s+k 2) X 1

! =(s2+2 s+1 ) X 2−( s+1 ) X 1(1)

F&

F

F&2

F2


15/23

fv 2+ fk 2−fk 1−fv1=m1 ´

1

v 2 ( ´ 2− ´ 1 )+k 2 ( 2− 1 )−k 1 1−v1 ´ 1=m1 ´ 1

0=(m1 s2+( v1+v2 ) s +k 2+k 1) X 1−(v2 s +k 2) X 2

0=(s2+3 s+2) X 1−( s+1 ) X 2

X 2=

s2+3 s +2

s+1 X

1(2)

Ree#"a+a%,' 27 e% .7:

! =(s2+2 s+1 ) s2

+3 s +2s+1 X 1−(s +1 ) X 1

X 1

! =

s+1( s2+2 s+1 ) ( s2+3 s+2 )− (s+1 )2

X 1

! =

1

s3+4 s2+4 s+1

8ara e" /i/te#a #e(;%i(' r'ta(i'%a" 3e /e #e/tra e% "a 4)ra e%(e%tre "a

%(i=% ,e tra%/ere%(ia G ( s )="

2( s)

T (s ) .

F&.

F.

F&2

F2


16/23

S'"(i=%:

# "́1=T −$

1"́1−$

2 ( "́1− "́2 )− % ( "1−"2 )

T =( s2+2 s+1 )"1−(s +1 )"2(1)

0=$2 ( "́1− "́2 )+ % ("1−"2)−$3 "́2

(s +1 )"1=(2 s+1 )"2

"1=

(2 s+1 )"2

s+1(2)

Ree#"a+a%,' 27 e% .7:

T =( s2+2 s+1 ) (2 s +1 )"

2

s+1−( s+1 ) "

2

2

.

T. T2"1

T2 T "2

T


17/23

"2

T =

s +1( s2+2 s +1) (2 s+1 )−(s+1)2

"2

T

= 1

2 s2

+2 s

B 8ara e" /i/te#a r'ta(i'%a" 3e /e #e/tra e% "a 4)ra1 e%(e%tre "a %(i=%

,e tra%/ere%(ia1 G ( s )=" L(s)/ T (s) *

S'"(i=%:

# "́2=T

2−T $10=T $1−T % 0=T % −T 30=T L−T $2

# "́2=T

2−$

1 ( "́2− "́3 ) $1 ( "́2− "́3 )= % ("3−"4 ) % ("3−"4 )=T 3 T L=$2"́ L

(# s2+$1 s)"2−$1 s "3=T 2 $1 s "2=( $1 s+ % )"3− % "4 T 3= % "3− % "4 T L=$2 s " L

A,e#;/ /ae#'/ 3e:

T

T 2= & 1

& 2= 10

100

T 3

T L=

" L

"4=

& 3

& 4=

50

10

T T2

"4"

3"

2

T. T. T T T T T2

T2


18/23

Le)' te%e#'/:

"4=0.2" L T L=0.2T 3T 2=10T

Re/'"&ie%,' e" /i/te#a ,e "a/ e(a(i'%e/ 3e e/t;% /ra$a,a/ e%('%tra#'/

"a %(i=% ,e tra%/ere%(ia*

"3=

5$2

s+0.2 %

% " L

"2=

($1 s+ % )(5 $2 s+0.2 % )−0.2 % 2

$1s%

" L

10T =

{(# s

2

+$1 s) [ ( $1 s+ % )( 5$2 s+0.2 % )−0.2 %

2

$1

s%

]−$1 s [

5$2

s+0.2 %

%

]}" L

Ree#"a+a%,' ,at'/:

s+¿(¿ (0.2s+0.2 )−0.2 ]−s2 (0.2s+0.2)

¿( s2+s )¿10T =¿

" L

T =

10s

0.2s4+0.4s3+0.2 s2

" L

T =

50

s3+2 s2+s

" L

T =

50

s (s +1 )2

9 Da,' e" /i/te#a ('#i%a,' tra/"a(i'%a" $ r'ta(i'%a" 3e /e #e/tra e% "a

4)ra1 e%(e%tre "a %(i=% ,e tra%/ere%(ia1 G ( s )= X (s)/T (s) .


19/23

S'"(i=%:

T −T % 1=# 1 "́

T =# 1

"́+ % 1("−"

1)

T =(# 1 s2+ %

1)"− % 1"1

T % 1−T ! −T ' 3=# 2 "́1

% 1 ("−"1 )−T ! − '3( "́1− "́2)=# 2 "́1

% 1

"+ '3

s "2=(# 2 s

2+ '3

s + % 1 )"1+ !r

T.

TF TD T.

TD


20/23

T '3=# 3 "́2

'3( "́

1− "́

2)=#

3"́2

'3

s "1=(# 3 s

2+ '3 s)"2

! − ! fv− ! k 2− ()= ( ́

! − ()= ( ´ +f v ´ + % 2

! =( ( s2+ f v s + % 2 ) X + ()

s

A,e#;/:

X ="1

r

Re/'"&ie%,' e" /i/te#a ,e e(a(i'%e/1 te%e#'/:

T =(# 1 s2+ % 1 )[ r (# 3 s2+ '3 s )[ X r ( # 2 s2+ '3 s + % 1 )+( ( s2+f v s+ % 2 ) Xr+ ()r ]−( ' 3 s)2 X

% 1r (# 3 s2+ '3 s)

]− % 1 X rT ={(# 1 s

2+ % 1 ) s [

r (# 3 s2+ '3 s) [(# 2 s

2+ '3 s+ % 1 )+( ( s2+ f v s+ % 2 )r

2

]−( '3 s)2 r

% 1r (# 3 s2+ '3 s) ]− % 1

1

r } X +(# 1 s

2+ % 1 % 1(# 3 s

2+

F

F& M) F2


21/23

.0 E" #'t'r ($a/ (ara(ter-/ti(a/ ,e ar ('%tra &e"'(i,a, /e #e/tra e% "a4)ra #e&e "a (ar)a 3e /e &e e% e" ,ia)ra#a* A")%'/ ,e "'/ e%)ra%a@e/

tie%e% i%er(ia* E%(e%tre "a %(i=% ,e tra%/ere%(ia1 G ( s )="2 (s )/ Ea (s ) .

De" )r;4(' te%e#'/:

%

Ra=

T

Ea=5

5=1

# 1

"́1=T m−f 2r1 # 2"́3=f 2 r2−T 2

# 1 s2 "1=T m− ! 2 r1 # 2 s2"3= ! 2 r2−T 2

C'#i%a%,' e(a(i'%e/ te%e#'/:

T 2=

T m−# 1 s2

"1

r1

r2−#

2s2

"3

T2

T#


22/23

A,e#;/ e% )e%era" /ae#'/ 3e:

r1

r2=

"2

"1=

& 1

& 2

Le)':

T 2=

T m−# 1 s2

"1

& 1

& 2−#

2s2

"3

T 2=

T m−s2

"1

1020−2s2"

3

T 2=2T m−2 s2

"1−2 s2

"3

#

(¿¿2+# 3) "́3=T 2−T 3¿

#

(¿¿2+# 3)s2

"3=T

2−T

3

¿

# 4

"́2=T

4−T '

# 4

"́2=T

4− ' "́

2

# 4

"́2=T

3

20

10− ' "́

2

T T2

T< TD


23/23

modelamiento de sistemas eléctricos y mecanicos

Documents