ÿþP o d s t a w y E l e k t r o t e c h n i k i - S t a n y n i e u s t a l o n e
I I . M e t o d a O p e r a t o r o w a
Z a d a n i e o . 1
W y z n a c z y p r d i 2 ( t ) .
P o z a s t o s o w a n i u p r z e k s z t a Bc e n i a L a p l a c e a
E
I o =
2 R
E
+ L i o
1 1 1 L i o
s
V ( + + ) = -
R + s L R + s L R + s L R + s L R + s L
E
+ L i o
3 L i o
s
V = -
R + s L R + s L R + s L
E
V =
s ( R + s L )
1
V E E
I 2 ( s ) = = =
R
R + s L 3 s ( R + s L )
3 L s ( + s )
L
R R
- t - t
E E E E
L L
i 2 ( t ) = + e = [ - e ]
3 R - 3 R 3 R 3 R
Z a d a n i e o . 2
O b l i c z y n a p i c i e n a z a c i s k a c h w y Bc z n i k a w c h w i l i t = 0 i n a p i c i a n a C 1 i C 2 w c h w i l i t = T .
a ) t = 0 :
E E
E C 3 E C 3 E 1
s s
J ( s ) = = = = =
1 2 R C 3 s + 1 1 1
2 R C 3 s + 1
2 R + ( s + ) 2 R C 3 2 R ( s + )
C 3 s C 3 s 2 R C 3 2 R C 3
1
- t
E
2 R C 3
i ( t ) = L - 1 { J ( s ) } = e
2 R
1 1
t t
- t - t
1 1 E 1 E
2 R C 3 2 R C 3
U ( t ) = e d t = [ - 2 R C 3 e ] t =
c 3 0
+"i ( t ) d t = C 3 +"
C 3 0 2 R C 3 2 R
0
1 1
- -
E
2 R C 3 t 2 R C 3
= [ - 2 R C 3 e + 2 R C 3 ] = E ( 1 - e
C 3 2 R
1 1
- t - t
E E
2 R C 3 2 R C 3
U ( t ) = i ( t ) R = R e = e
R
2 R 2
1 1 1 1
- - t - t - t
E 1
2 R C 3 2 R C 3 2 R C 3 2 R C 3
U w y l ( t ) = U c 3 ( t ) + U R ( t ) = E ( 1 - e ) + e = E ( 1 - e + e )
2 2
b ) t = T
E
E ( C 1 + C 2 ) E ( C 1 + C 2 ) E 1
s
J ( s ) = = = =
1 1 1
R s ( C 1 + C 2 ) + 1 R
R + ( s + s +
s ( C 1 + C 2 ) R C ( C 1 + C 2 ) R ( C 1 + C 2 ) R C ( C 1 + C 2 )
1
- t
E
R ( C 1 + C 2 )
J ( t ) = L - 1 { J ( s ) } = e
R
O b l i c z a m y n a p i c i e n a C 1 i C 2 p a m i t a j c |e : C z = C 1 + C 2
2
1 1
t t
- t - t
1 1 E E
R ( C 1 + C 2 ) R C ( C 1 + C 2 )
U c 1 ( t ) = U c 2 ( t ) = e d t = [ - R C ( C 1 + C 2 ) e ] t =
0
+"i ( t ) d t = C z +"
C z 0 R R C z
0
1 1
- t - t
- E R ( C 1 + C 2 ) E R ( C 1 + C 2 )
R ( C 1 + C 2 ) R ( C 1 + C 2 )
e + = E ( 1 - e )
R ( C 1 + C 2 ) R ( C 1 + C 2 )
Z a d a n i e o . 3
W y z n a c z y r o z p By w p r d ó w w u k Ba d z i e p r z e d s t a w i o n y m p o n i |e j d l a d w ó c h p r z y p a d k ó w
z a m y k a n i a i o t w i e r a n i a w y Bc z n i k a .
D l a o b w o d u p r z e d s t a w i o n e g o n a r y s u n k u r ó w n a n i a r ó |n i c z k o w e m a j p o s t a :
d i 1 d i 2
R 1 i 1 + L 1 - M = u 1
d t d t
d i 1 d i 2
- M + R 2 i 2 + L 2 u = 0
d t d t
T e s a m e r ó w n a n i a w r a c h u n k u o p e r a t o r o w y m :
( R 1 + s L 1 ) I 1 ( s ) - s M I 2 ( s ) = U 1 ( s ) + L 1 i 1 ( 0 ) - M i 2 ( 0 )
- s M I 1 ( s ) + ( R 2 + s L 2 ) I 2 ( s ) = L 2 i 2 ( 0 ) - M i 1 ( 0 )
P r z y z e r o w y c h w a r u n k a c h p o c z t k o w y c h r ó w n a n i a p o w y |s z e u p r a s z c z a j s i d o p o s t a c i
Z 1 ( s ) I 1 ( s ) - s M I 2 ( s ) = U 1 ( s )
- s M I 1 ( s ) + Z 2 ( s ) I 2 ( s ) = 0
p r z y c z y m : Z 1 ( s ) i m p e d a n c j a o p e r a t o r o w a o b w o d u 1 ;
Z 2 ( s ) i m p e d a n c j a o p e r a t o r o w a o b w o d u 2 .
1 . Z a m y k a n i e w y Bc z n i k a .
U
Z r ó w n a D p r z y U ( s ) = o t r z y m u j e m y
S
Z 2 ( s ) U
I 1 ( s ) =
2
Z 1 ( s ) Z 2 ( s ) - s 2 M s
M
I 2 ( s ) = U
2
Z 1 ( s ) Z 2 ( s ) - s 2 M
3
P o p o d s t a w i e n i u : Z 1 ( s ) = R 1 + s L 1 ; Z 2 ( s ) = R 2 + s L 2 ; L 1 L 2 = M 2 o t r z y m u j e m y
s L 2 + R 2
I 1 ( s ) = U
s [ s ( L 1 R 2 + L 2 R 1 ) + R 1 R 2 ]
R 2
s +
U L 2 L 2
I 1 ( s ) =
L 1 R 2 + L 2 R 1 s [ s R 1 R 2 ]
+
L 1 R 2 + L 2 R 1
R o z k Ba d a m y n a u Ba m k i p r o s t e
U L 2 L 1 R 2 + L 2 R 1 1 L 1 R 2 1
I 1 ( s ) = [ - ]
L 1 R 2 + L 2 R 1 L 2 R 1 s L 2 R 1 s + R 1 R 2
L 1 R 2 + L 2 R 1
U 1 L 1 R 2 1
I 1 ( s ) = [ -
R 1 s L 1 R 2 + L 2 R 1 s + R 1 R 2
L 1 R 2 + L 2 R 1
L 1 L 2
P o p o d s t a w i e n i u = Ä1 , = Ä2 o t r z y m u j e m y p r z e b i e g p r d u
R 1 R 2
t
U Ä1 - Ä1 + Ä2
i ( t ) = [ 1 - e ]
R 1 Ä1 + Ä
2
P r d w o b w o d z i e 2
M U
I 2 ( s ) =
s ( L 1 R 2 + L 2 R 1 ) + R 1 R 2
M U 1
I 2 ( s ) =
L 1 R 2 + L 2 R 1 s + R 1 R 2
L 1 R 2 + L 2 R 1
t
-
M U
Ä1 + Ä2
i 2 ( t ) = e
L 1 R 2 + L 2 R 1
P r d y w s t a n i e u s t a l o n y m :
U
i 1 u = ; i 2 u = 0
R 1
2 . O t w i e r a n i e w y Bc z n i k a .
4
P o o t w a r c i u w y Bc z n i k a o b w ó d 1 z o s t a j e p r z e r w a n y , w i c i 1 = 0 . P r d m o |e p By n t y l k o w
o b w o d z i e 2 . W c e l u j e g o o b l i c z e n i a p o d s t a w i m y w r ó w n a n i a c h :
U
I 1 ( s ) = 0 ; i 1 ( 0 - ) = ; i 2 ( 0 - ) = 0 p r z y j m u j c , |e p r z y z a m k n i t y m w y Bc z n i k u z o s t a B u p r z e d n i o
R 1
o s i g n i t y s t a n u s t a l o n y .
M U
I 2 ( s ) ( s L 2 + R 2 ) = - M i 1 ( 0 ) = -
R 1
M U 1
I 2 ( s ) = -
L 2 R 1 s + R 2
L 2
s t d
t
-
M U
Ä2
i 2 ( t ) = - e
L 2 R 1
Z a d a n i e o . 4
W o b w o d z i e j a k n a r y s u n k u w c h w i l i t = 0 z a m k n i t o w y Bc z n i k W . O b l i c z y p r z e b i e g i n a p i
n a o b y d w u k o n d e n s a t o r a c h , j e |e l i w i e m y , |e u 1 ( 0 ) = u 2 ( 0 ) = 0 .
T = 0
R
R
W
C
C
E = c o n s t
U 1 ( t )
U 2 ( t )
R o z w i z a n i e :
K o r z y s t a j c z m e t o d y p o t e n c j a Bó w w z Bo w y c h o t r z y m u j e m y :
5
E
p R E ( R C p + 1 )
U 1 ( p ) = =
2
1 1
p [ ( R C ) 2 p + 3 R C p + 1 ]
+ p C +
1
R
R +
p C
U 1 ( p ) 1 E
U ( p ) = Å" =
2
1
p C p [ ( R C ) 2 p 2 + 3 R C p + 1 ]
R +
p C
P r z y p r z e j [c i u n a p o s t a c z a s o w k o r z y s t a m y z I w z o r u H e a v i s i d e a .
R ó w n a n i e m a p o s t a : H ( p ) = ( R C ) 2 p 2 + 3 R C p + 1 = 0 p o s i a d a d w a n a s t p u j c e p i e r w i a s t k i :
3 3 1 1
p 1 , 2 = - ± ( ) 2 - ( ) 2 = ( - 1 , 5 ± 1 , 1 1 8 )
2 R C 2 R C R C R C
0 , 3 8 2 2 , 6 1 8
p 1 = - p 2 = -
R C R C
P o o b l i c z e n i a c h m a m y :
R C p 1 + 1 R C p 2 + 1
1 2
u 1 ( t ) = E Å"[ 1 + e p t + e p t ] =
( R C ) 2 p 1 ( p 1 - p 2 ) ( R C ) 2 p 2 ( p 2 - p 1 )
0 , 3 2 8 t 2 , 6 1 8 t
- -
R C R C
= E Å"[ 1 - 0 , 7 2 3 e - 0 , 2 7 6 e ] Å"1 ( t )
p 1 t p 2 t
e e
u 2 ( t ) = E Å"[ 1 + + ] =
( R C ) 2 p 1 ( p 1 - p 2 ) ( R C ) 2 p 2 ( p 2 - p 1 )
0 , 3 8 2 t 2 , 6 1 8 t
- -
R C R C
= E Å"[ 1 - 1 , 1 7 0 Å" e + 0 , 1 7 0 Å" e ] Å"1 ( t )
Z a d a n i e o . 5
W y z n a c z y n a p i c i a n a k o n d e n s a t o r z e .
6
ëø öø
1 E E E
ìø ÷ø
I 1 ( s ) ìø R + - I 2 ( R ) = - =
s C 1 ÷ø s 2 s 2 s
íø øø
ëø öø
1
ìø ÷ø
I 2 ( s ) ìø R + R + - I 1 ( R ) = 0
s C 2 ÷ø
íø øø
C 1
I 1 ( s ) ( R s C 1 + 1 ) - I 2 ( s ) R s C 1 = E
2
I 2 ( s )
- I 1 ( s ) R s C 2 + ( 2 R s C 2 + 1 ) = 0
2
R s C 1 + 1 - R s C 1
W g = = R 2 C 1 C 2 s 2 + s ( R C 1 + 2 R C 2 ) + 1
- R s C 2 2 R s C 2 + 1
C 1
E - R s C 1 = E C 1 ( 2 R s C 2 + 1 )
W I =
2
1
2
0 2 R s C 2 + 1
C 1 C 1
R s C 1 + 1 E
W I = = E R s C 2
2
2
2
- R s C 2 0
E
C 1 C 2 R s
2
I 2 ( s ) =
R 2 C 1 C 2 s 2 + ( R C 1 + 2 R C 2 ) s + 1
E
C 1 R
2
U C 2 ( s ) =
R 2 C 1 C 2 s 2 + ( R C 1 + 2 R C 2 ) s + 1
C 1 = C = C
2
2
M ( s ) = s 2 C R 2 + s 3 R C + 1
2
› = 9 ( R C ) 2 - 4 C R 2 = 5 ( R C ) 2
- 3 5
r 1 , 2 = ±
2 R C 2 R C
- 0 . 7 6 - 0 . 3 8
s 1 = =
2 R C R C
- 3 Å" 2 . 2 4 - 2 . 6 2
s 2 = =
2 R C R C
7
E
ëø öø
ìø ÷ø
E 1 1
2
1 2
ìø
U C 2 ( s ) = E" e r t + e r t ÷ø
0 . 3 8 2 . 6 2 2 . 6 2 0 . 3 8
R C ( s - s 1 ) ( s - s 2 ) 2 R C ìø ÷ø
- + - +
ìø ÷ø
íø R C R C R C R C øø
ëø öø
ìø ÷ø
E 1 1
1 2
ìø
U C 2 ( t ) = e r t + e r t ÷ø
2 . 2 4 2 . 2 4
2 R C ìø ÷ø
-
ìø ÷ø
íø R C R C øø
E 1 E
ëø
1 2 1 2
U C 2 ( t ) = ( e r t - e r t ) öø = ( e r t - e r t )
ìø ÷ø
2 2 . 2 4 4 . 4 8
íø øø
Z a d a n i e o . 6
W y z n a c z y u c ( t ) d l a w y m u s z e n i a p r z e d s t a w i o n e g o n a r y s . o . 6
R y s . o . 6 P r z e b i e g e ( t ) i j e g o r o z k Ba d n a s k Ba d o w e e 1 , e 2 .
P r z e b i e g e ( t ) m o |n a p r z e d s t a w i j a k o s u m p r z e b i e g ó w e 1 ( t ) i e 2 ( t ) .
8
E
e 1 ( t ) = t
Ä1
E
e 2 ( t ) = - ( t - Ä1 )
Ä1
E E
e ( t ) = t - ( t - Ä1 )
Ä1 Ä1
P r z e c h o d z c n a p o s t a o p e r a t o r o w :
E E E
1 1
E ( s ) = - e - s Ä = [ 1 - e - s Ä ]
Ä1 s 2 Ä1 s 2 Ä1 s 2
N a p i c i e n a k o n d e n s a t o r z e w y z n a c z a m y n a s t p u j c o :
E ( s ) = I ( s ) R + U ( s ) Ò! E ( s ) - U ( s ) = I ( s ) R
1 E ( s ) - U ( s ) * 1 1 1
U ( s ) = I ( s ) * = = E ( s ) * - U ( s ) *
s C R s C s R C s R C
1 1
ëø1 öø
U ( s ) * + = E ( s ) *
ìø ÷ø
s R C s R C
íø øø
U ( s ) * ( s R C + 1 ) = E ( s )
E ( s ) E ( s )
U ( s ) = =
1 + s R C 1
öø
R C ëø s +
ìø ÷ø
R C
íø øø
E
îø ùø
1
( 1 - e - s Ä )
ïø úø
1
Ä1 s 2 E ( 1 - e - s Ä ) E 1 1
1
ïø úø
U ( s ) = = = - e - s Ä
1 1 Ä1 R C 1 1
öø öø ïøs ëøs + öø ëø öø úø
2
R C ëø s + Ä1 s 2 R C ëø s + s 2 s +
ìø ÷ø ìø ÷ø ìø ÷ø ìø ÷ø
ïø úø
R C R C R C R C
íø øø íø øø íø øø íø øø
ðø ûø
O p i e r a j c s i n a z a l e |n o [c i :
a 2
a" a t - ( 1 - e - a t )
s 2 ( s + a )
O t r z y m u j e m y :
9
2
1 îø ùø
E ëø öø ïø
ìø ÷ø
úø
R C 1 1
íø øø
ïø
U ( s ) = - e - s Ä1 úø =
2
1 1
ïø ëø öø ëø öø úø
1 2 2
ëø öø
s s + s s +
ìø ÷ø ìø ÷ø
Ä R C
ìø ÷ø
1 ïø úø
R C R C
íø øø íø øø
R C ðø ûø
íø øø
2 2
îø ùø
1 1
ëø öø ëø öø
ïø ìø ÷ø ìø ÷ø úø
E R C R C
íø øø íø øø
ïø 1
= - e - s Ä úø
1 ïø úø
1 1
ëø öø ëø öø
2 2
Ä
s s + s s +
ìø ÷ø ìø ÷ø
1
ïø úø
R C
R C R C
íø øø íø øø
ïø úø
ðø ûø
P r z e c h o d z c n a p o s t a c z a s o w o t r z y m u j e m y :
1 1
îø ùø
îø ùø îø ùø
ëø - t öø ëø - ( t - Ä1 ) öø
E R C 1
ìø1 - R C ÷øúø 1 ìø1 - R C ÷øúøúø
( t
U ( t ) = ïø t - e - ïø - Ä1 ) - e
ïø
ìø ÷øúø ïø R C ìø ÷øúøúø
Ä1 ïøðø R C
øø íø øø
ûø ðø ûø
ðøïø íø ûø
Z a d a n i e o . 7
e ( t ) = E + E ( t - Ä) - 2 E ( t - 2 Ä)
E E 2 E
E ( s ) = + e - Äs - e - 2 Äs
s s s
1 0
E ( s ) = I ( s ) R + I ( s ) S L
E ( s ) = I ( s ) ( R + S L )
E ( s )
= I ( s )
R
L ( + S )
L
1 1 2
E E 1 e - Äs 2 e - 2 Äs
S S S
I ( s ) = [ + e - Äs - e - 2 Äs ] = [ + - ]
R R R R R R
L L
( + S ) ( + S ) ( + S ) S ( S + ) S ( S + ) S ( S + )
L L L L L L
K o r z y s t a j c z z a s a d y :
F 1 ( s )
F 1 ( s k )
f ( t ) = = £ e s t
F 2 ( s ) F 2 ' ( s k )
o t r z y m u j e m y :
ëø öø
R
R R
ìø ÷ø
- t - ( t - Ä ) - ( t - 2 Ä )
E 1 1 1 1 1 1
L L L
i ( t ) = [ ( + e ) + ( + e ) - ìø ÷ø
+ e ) ]
R R R R R R
L
ìø ÷ø
-
- -
ìø ÷ø
íø L L øø
L L L L
R R R R R R
- t - ( t - Ä ) - ( t - 2 Ä ) - t - ( t - Ä ) - ( t - 2 Ä )
E E
L L L L L L
i ( t ) = [ ( 1 - e ) + ( 1 - e ) - ( 1 - e ] = [ 1 - e - e + e ]
R R
Z a d a n i e o . 8
W o b w o d z i e p r z e d s t a w i o n y m n a r y s u n k u o b l i c z y n a p i c i e n a c e w c e .
1 1
N a p i c i e z a s i l a j c e m a p o s t a :
P o s z c z e g ó l n e p r z e b i e g i w y n o s z ;
E
e 1 ( t ) = t Å"1 ( t )
T
E
e 2 ( t ) = - t Å"1 ( t - T )
T
E
e 3 ( t ) = t Å"1 ( t - 2 T )
T
E
e 4 ( t ) = - t Å"1 ( t - 3 T )
T
e 5 ( t ) = - 2 E Å"1 ( t - 4 T )
C a Bk o w i t y p r z e b i e g n a p i c i a
e ( t ) = e 1 ( t ) + e 2 ( t ) + e 3 ( t ) + e 4 ( t ) + e 5 ( t )
E E E E
e ( t ) = t Å"1 ( t ) - t Å"1 ( t - T ) + t Å"1 ( t - 2 T ) - t Å"1 ( t - 3 T ) - 2 E Å"1 ( t - 4 T )
T T T T
c z y l i
E 1 E 1 E 1 E 1 1
E ( s ) = Å" - Å" e - s T + Å" e - s 2 T - Å" e - s 3 T - 2 E e - s 4 T
T T T T s
s 2 s 2 s 2 s 2
E ( s )
I ( s ) =
( 3 R + s L )
T y m s a m y m n a p i c i e U L :
E ( s )
U L ( s ) = Å"s L
( 3 R + s L )
P o p o d s t a w i e n i u d o w z o r u n a n a p i c i e
1 2
E 1 E 1 E 1 E 1 1
Å" - Å" e - s T + Å" e - s 2 T - Å" e - s 3 T - 2 E e - s 4 T
T T T T s
s 2 s 2 s 2 s 2
U L ( s ) = Å" s L
3 R + s L
E L E L e - s T E L e - s 2 T E L e - s 3 T 2 E L e - s 4 T
U L ( s ) = - + - +
T s ( 3 R + s L ) T s ( 3 R + s L ) T s ( 3 R + s L ) T s ( 3 R + s L ) ( 3 R + s L )
O b l i c z a m y p o s z c z e g ó l n e c z Bo n y
E L E L 1
U 1 ( s ) = = Å"
L
3 R
T s ( 3 R + s L ) T L
s ( s + )
L
3 R 3 R
- t - t
E L E
L L
u 1 ( t ) = Å" ( 1 - e ) = ( 1 - e )
L
T L 3 R T 3 R
E L e - s T E L e - s T
U 2 ( s ) = - = Å"
L
3 R
T s ( 3 R + s L ) T L
s ( s + )
L
3 R 3 R
îø ùø îø ùø
- ( t - T ) - ( t - T )
ïø úø ïø úø
E L E
L L
u 2 ( t ) = Å" Å"1 ( t - T ) úø = Å"1 ( t - T ) úø
ïø1 ( t - T ) - e ïø1 ( t - T ) - e
L
T L 3 R T 3 R
ïø úø ïø úø
ðø ûø ðø ûø
E L e - s 2 T E L e - s 2 T
3
U L ( s ) = = Å"
3 R
T s ( 3 R + s L ) T L
s ( s + )
L
3 R 3 R
îø ùø îø ùø
- ( t - 2 T ) - ( t - 2 T )
ïø úø ïø úø
E L E
3
L L
u L ( t ) = Å" Å"1 ( t - 2 T ) úø = Å"1 ( t - 2 T ) úø
ïø1 ( t - 2 T ) - e ïø1 ( t - 2 T ) - e
T L 3 R T 3 R
ïø úø ïø úø
ðø ûø ðø ûø
E L e - s 3 T E L e - s 3 T
U 4 ( s ) = - = Å"
L
3 R
T s ( 3 R + s L ) T L
s ( s + )
L
3 R 3 R
îø ùø îø ùø
- ( t - 3 T ) - ( t - 3 T )
E L ïø úø E ïø úø
u 4 ( t ) = Å" Å"1 ( t - 3 T ) úø = Å"1 ( t - 3 T ) úø
ïø1 ( t - 3 T ) - e L ïø1 ( t - 3 T ) - e L
L
T L 3 R T 3 R
ïø úø ïø úø
ðø ûø ðø ûø
1 3
2 E L e - s 4 T e - s 4 T
U 5 ( s ) = = - 2 E Å"
L
3 R
( 3 R + s L )
( s + )
L
3 R
îø ùø
- ( t - 4 T )
ïø úø
L
u 5 ( t ) = - 2 E ïøe Å"1 ( t - 4 T ) úø
L
ïø úø
ðø ûø
N a p i c i e n a c e w c e w y n o s i
2
u L ( t ) = u 1 ( t ) + u L ( t ) + u 3 ( t ) + u 4 ( t ) + u 5 ( t )
L L L L
ñø 3 R 3 R üø
îø ùø
- t - ( t - T )
ôø ôø
ïø úø
E
L
u ( t ) = Å"1 ( t - T ) úø + ýø
òø( 1 - e L ) - ïø1 ( t - T ) - e
T 3 R
ôø ôø
ïø úø
ðø ûø
óø þø
3 R 3 R
îø ùø îø ùø
- ( t - 2 T ) - ( t - 3 T )
ïø úø ïø úø
L L
+ - 2 T ) - e Å"1 ( t - 2 T ) úø - - 3 T ) - e Å"1 ( t - 3 T ) úø +
ïø1 ( t ïø1 ( t
ïø úø ïø úø
ðø ûø ðø ûø
3 R
ñø üø
- ( t - 4 T )
ôø ôø
L
- 2 E òøe Å"1 ( t - 4 T ) ýø
ôø ôø
óø þø
Z a d a n i e o . 9
O b l i c z y p r z e b i e g p r d u w o b w o d z i e j e |e l i w u k Ba d z i e d z i a Ba w y m u s z e n i e o p r z e b i e g u j a k n a
r y s u n k u :
R o z k Ba d a m y p o d a n f u n k c j n a s k Ba d o w e :
T 3
e ( t ) = E 1 ( t ) - E 1 ( t - ) + E 1 ( t - T ) - E 1 ( t - T ) + E 1 ( t - 2 T ) - . . .
2 2
d o k o n u j e m y p r z e k s z t a Bc e n i a L a p l a c e a
1 4
T 3
E ( s ) = L [ E 1 ( t ) - E 1 ( t - ) + E 1 ( t - T ) - E 1 ( t - T ) + E 1 ( t - 2 T ) - . . . ]
2 2
T 3
- s - s T
E E E E E
2 2
E ( s ) = - e + e - s T - e + e - 2 T - . . .
s s s s s
s T 3
- - s T
E
2 2
E ( s ) = ( 1 - e + e - s T - e + e - 2 s T - . . . )
s
W y r a |e n i e w n a w i a s i e j e s t c i g i e m g e o m e t r y c z n y m w i c :
E 1
E ( s ) = ( )
s T
s -
2
1 + e
O b l i c z a m y p r d w o b w o d z i e :
T 3
- s - s T
E
s T 3
2 2
- - s T
C s ( 1 - e + e - s T - e + . . . )
2 2
E ( s ) E ( s ) * C s E 1 - e + e - s T - e + . . .
s
J ( s ) = = = = *
1 1 1
R s C + 1 s
R + ( s + ) R C ( s + )
s C R C R C
D o k o n u j e m y o d w r o t n e g o p r z e k s z t a Bc e n i a L a p l a c e a
1 1 T 1
- t - ( t - ) - ( t - T )
E T
R C R C 2 R C
i ( t ) = L - 1 { J ( s ) } = * [ e * 1 ( t ) - e * 1 ( t - ) + e * 1 ( t - T ) . . . ]
R 2
Z a d a n i e o . 1 0
O k r e [l i o d p o w i e d z u ( t ) u k Ba d u R L n a n a p i c i e w y m u s z a j c e e ( t ) j a k n a r y s u n k u
R
e ( t )
L U L
e ( t )
2 E
E
t
T
1 5
Z a d a n i e o . 1 1
W y z n a c z y n a p i c i e n a k o n d e n s a t o r z e d l a w y m u s z e n i a p r z e d s t a w i o n e g o p o n i |e j .
1 6
T
öø
e ( t ) = E m Å" s i n Ét Å"1 ( t ) - 2 E m Å" s i n Ét Å"1 ëøt - ÷ø
+ E m Å" s i n Ét Å"1 ( t - T )
ìø
2
íø øø
E ( s ) 1
U C ( s ) =
1
s C
R +
s C
E ( s )
U C ( s ) =
1 + s R C
îø T T ùø T
ëø öø öø
e ( t ) = E m Å" s i n Ét Å"1 ( t ) - 2 E m Å" s i n - ÷ø ìø
Å"1 ëøt - ÷ø
+ E m Å" s i n [ É( t - T ) + ÉT ] Å"1 ( t - T )
ïøÉìøt 2 + É 2 úø
2
íø øø íø øø
ðø ûø
ÉT T
j - s
1 1 1
j ÉT
2 2
E ( s ) = E m - 2 E m e Å" e Å" + E m e Å" e T s Å"
s - j É s - j É s - j É
2
g d z i e : É =
T
T
ëø
E ( s ) 1 E m ìø1 j - s öø
1
2
U C ( s ) = Å" = - 2 e Å" e + e 2 - T s ÷ø Å"
ìø ÷ø
1
s C R C 1
öø
íø øø
R +
( s - j É) ëø s +
ìø ÷ø
s C
R C
íø øø
1 T T 1
ëø
t öø ëø j É - t öø
E m 1 T
öø
j Ét
ìøëø - 2 e j Ét - e - R C 2 2 R C R C
ìø ÷ø ÷ø
U C ( t ) = Å"1 ( t ) + 2 ìøe Å" e - e Å" e Å"1 ëøt +
ìø ÷ø
ìø ÷ø ìø ÷ø
ìøíø
1
R C 2
íø øø
øø íø øø
íø
+ j É
R C
T 1
ëø - t öø
j ÉT j Ét
R C R C
ìø ÷ø
+ e Å" e - e Å" e Å"1 ( t - T ) )
ìø ÷ø
íø øø
Z a d a n i e o . 1 2
W y z n a c z y U L ( t ) d l a n a p i c i a z a s i l a n i a e ( t ) j a k n a w y k r e s i e .
1 7
N a p i c i e w y m u s z a j c e :
T
e ( t ) = E m s i n Ét 1 ( t ) + E m s i n Ét 1 ( t - )
2
T
- s
1 1
2
E ( s ) = E m + E m e
s - j É s - j É
O d p o w i e d z u k Ba d u :
T
ëø - s öø
E ( s ) s
2
÷ø
U ( s ) = s L = E m ìø1 + e
L
ìø ÷ø
R
R + s L
íø øø
( s - j É) ( s + )
L
s A B
= +
R - a s + b
s
( s - j É) ( s + )
L
A ( s + b ) + B ( s - a ) s
=
( s - a ) ( s + b ) ( s - a ) ( s + b )
A s + B s + A b - B a = s
A + B = 1
ñø
òø
óøA b - B a = 0
A = 1 - B
b ( 1 - B ) - B a = 0
b - B ( b + a ) = 0
b
B =
b + a
a
A =
b + a
1 îø ùø
îø1 ( t ) + 1 ( t - T
U ( t ) = E m ) ùø Å" ( A e - a t + B e b t ) úø
L
ïø
ïø úø
R
2
ðø ûø
ðø ûø
( + j É)
L
P o p o d s t a w i e n i a c h o t r z y m a m y :
ÉL ÉL R
îø j ( Ét - a r c t g ) - j a r c t g - t ùø
E m
R R L
U ( t ) = + R e e
ïøÉL e úø1 ( t ) +
L
( R 2 + ( ÉL ) 2
ðø ûø
ÉL ÉL R T R
îø j ( Ét - a r c t g ) - a r c t g - t ùø
E m T
R R 2 L L
+ + R e e e - )
ïøÉL e úø1 ( t 2
R 2 + ( ÉL ) 2
ðø ûø
P r z y j m i j m y o z n a c z e n i e :
ÉL
Õ = a r c t g
R
R
- t
L
u L ( t ) = ( E m s i n Õ c o s ( Ét - Õ) - E m c o s Õ s i n Õe ) 1 ( t ) +
R T R
- t
T
2 L L
+ ( - E m s i n Õ c o s ( Ét - Õ) - E m c o s Õ s i n Õe e ) 1 ( t - )
2
O s t a t e c z n i e :
1 8
ñø
ôø 0 d l a t < 0
R
ôø
- t
ôø 1 T
L
u L ( t ) = s i n Õ c o s ( Ét - Õ) - E m s i n 2 Õe d l a 0 < t <
òøE
m
2 2
ôø
R T R
- t
1 T
ôø
2 L L
- E m s i n 2 Õ( 1 + e ) e d l a t >
ôø
óø 2 2
Z a d a n i e o . 1 3
W y z n a c z y i ( t ) d l a n a p i c i a z a s i l a n i a e ( t ) j a k n a w y k r e s i e .
F u n k c j n a p i c i a e ( t ) m o |n a r o z Bo |y n a t r z y s k Ba d o w e :
1 9
E 2 E E
e ( t ) = t - ( t - Ä ) + ( t - 2 Ä )
Ä Ä Ä
E 1 2 E 1 E 1 E 1
e ( s ) = - e - s Ä + e - 2 s Ä = [ 1 - 2 e - s Ä + e - 2 s Ä ]
Ä s 2 Ä s 2 Ä s 2 Ä s 2
P r d w o b w o d z i e o b l i c z a m y z :
e ( s ) e ( s ) E ( 1 - 2 e - s Ä + e - 2 s Ä )
I ( s ) = = =
R R
R + s L
L ( s + ) s 2 ÄL ( s + )
L L
1 R
S z u k a m y o b r a z u f u n k c j i g d z i e a =
s 2 ( s + a ) L
1 A s + B C ( A s + B ) ( s + a ) + C ( s ) 2 A s 2 + B s + A a s + B a + C s 2
= + = =
s 2 ( s + a ) s 2 s + a s 2 ( s + a ) s 2 ( s + a )
P o r ó w n u j e m y w s p ó Bc z y n n i k i :
A + C = 0
B + A a = 0
1 - 1 1
a B = 1 B = A = C =
2 2
a a a
P o d s t a w i a m y A B C
1 1 1
a 2 s a a 2 ëø 1 1 öø ëø 1 1 öø ëø 1 1 öø
- + + = * + * + *
ìø ÷ø ìø ÷ø ìø ÷ø
s 2 s 2 s + a a 2 s a s 2 øø íø a 2 s + a
íø øø íø øø
P r z e c h o d z c n a p o s t a c z a s o w :
1 1 1
1 ( t ) + t + e - a t
a 2 a a 2
P o d s t a w i a j c
îø ùø
ëø öø ëø öø
R R
÷ø ìø ÷ø
ïøìø 1 úø
1 1 - t
1 1 1 - ( t - Ä )
ïøìø - R 1 ( t ) + R t + R e L ÷ø * 1 ( t ) - 2 ìø - R 1 ( t - Ä ) + R 1 ( t - Ä ) + R e L ÷ø * 1 ( t - Ä ) úø
ìø
ïøìø ( ) 2 úø
( ) 2 ÷ø ( ) 2 ( ) 2 ÷ø
ìø ÷ø ìø ÷ø
E L L L øø íø L L L øø úø
íø
ïø
I ( t ) =
ïø úø
ÄL
ëø öø
ïø úø
R
ìø ÷ø
( t - 2 Ä )
ïø+ ìø 1 1 ( t - 2 Ä ) + 1 1 ( t - 2 Ä ) + 1 e - L úø
÷ø - 2 Ä )
* 1 ( t
R R R
ïø úø
ìø ÷ø
( ) 2 ( ) 2
ìø ÷ø
ïø úø
íø L L L øø
ðø ûø
2 0
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