א٢٦٢
א٢٦٢ EF
א
W،،אא،א
א א א אא א א א א
אאא،אאאאאא אאאאא
אWאאאאאא K אא
אאאאאאא
א ،א אא
א א א א א א א ،
א،אאאא
אאאאאאא אאאאא אאא،א
Kאא،א
?? ?אא
Kאאאאאאאא? אאאאאאא
،א،אאאא Kאאאאא
W א א א א
Kא
אאאא
א٢٦٢ EF
א
Wאא Wאאא Kאא
K١
Kאאאא
K٣
Kאא Kא
Kאא
K٢
K٤ K٥
Wאא Wא
Kא
K١
Kא
K٣
K
Kאאא
Kאא
Kאא
K٢
K٤ K٥ K٦
K٣٨Wאא KאWאא
EF
א
א
١
אא
א٢٦٢
א
א
EF
Wאא Wא
Kאא
K١
Kאאא
K٣
Kאאאאא
K٥
Kאאאא Kאא
Kאאאא
Kאאאאא
-١-
K٢
K٤
K٦
K٧
אא
א٢٦٢
א
EF
א
W ١ J١ א אאאאאאאאא אא، אאאK אא
אא،אאא
אאאאא אא،א אא،א،אאאא KאKKKK אאאאא
אאאאאא
אKאאאאאא אאאא
Kא
אאאאאא
، אאא אאא אאאא
אאאאא
Kאאאאאא
אאאאאאאאא
Kאאאא
אא אא
אאא،א
אאאאאאא
KאKKאאאאאאא
Wאאא ٢ J١ אאאאא،אא אא،אאאאא
א، אאאא
אאאאאאא، -٢-
אא
א٢٦٢
א
EF
א
،א،
אK אאאאא
א א אאא
אאאאא Kא
אאאא
אאאKאא אאאאאאא
Kאא
Wאא ٣ J١ א אאא ،אאאאאאאא
אאאאא אאאאאא א
K אא ،אאאאא
אאא
KאKKKK
Wאאאאא א٤ J١ Wאאאא
، אאא א ، אאאא Kאאא
E
אאא E א، אא، א
Kאאאאא،א
-٣-
אא
א٢٦٢
א
א
EF
،E אאFאאאא
אEאFאאא،א Kאאאא
אאאאאא
אאאאאא
E
E
אא
אאאאאא
KEF
Wאא ٥ J١ E אFאאאE١ J١Fא א، א Kא
א
א א
אאא
א
H
J
א
א
אאא
אא א
א א
אא
א א
אא
Kאאאאא،E١ J١F
Wא
א א א Kא
E
Wא אE
אאאא
Kאאא
-٤-
אא
א٢٦٢
א
א
EF
Wא
אאאאאא
KאKEאFאא Wא
אאאא، א
Kאאאא
E
E
Wאא
E
Wאא
E
Kאאאא א אאא אאא
K
Wא
אאאא
E
אא، אא
Kאא Wאא
א אאEאF א
Kא
E
Wאא אE
אאא
،אא א Kאאא
-٥-
אא
א٢٦٢
א
EF
א
W אאאאא، אא K١ ؟אאאאKא
אאאא Kאאאאאאאא ؟אאא
؟אא،אאא
؟אאאא
Wא
Kאא
E
Kא
E
Kאא
E
K אE
Kאא
-٦-
E
K٢
K٣
K٤
K٥
EF
٢
אא
א٢٦٢
א
EF
Wאא Wא
K
K١
K
K٣
Kא
K٥
K
Kאאא
Kאאא
-٧-
K٢ K٤
K٦
אא
א٢٦٢
א
EF
W ١ J٢ אאאאאאאא
אאאאKא אאא،א
אאאאK ، K
W ٢ J٢ אאאא
FאאK א
KאאEא Jא W ∞
F ( s) = L( f (t )) = ∫ f (t )e− st dt 0
KE s = σ +
W
jω FאאאאאWF(s)
E
KWL
E
KאאאאWf(t) E
Wא L−1 ( F ( s )) = L−1 ( L( f ( s ))) = f ( s )
c + j∞
=
1 F ( s )e st ds 2πj c −∫j∞
אאאאאא
Kאאאאאאא،א
W ٣ J٢ אאא
א، אאאא WאKאאאאא -٨-
אא
א٢٦٢
א
EF
Wא
E
L( af (t ) + bg (t )) = aF ( s ) + bG ( s )
Wא E L( Af (t )) = AL( f (t )) L( f (t ) ± g (t )) = F (s) ± G (s)
Wאא
E
Wאא
E
n f (t ) ⎞ n ⎟⎟ = s F ( s ) − ∑ k =1 s n − k f ( k −1) (0) n ⎝ dt ⎠
⎛
L⎜⎜ d
n
WKאאאf(0) f ( k −1) (t ) = d
k −1
dt
f (t )
k −1
( f (t )dt )t =0 L(∫ f (t )dt ) = F ( s) + ∫ s s
(
)
L ∫∫ f (t )dtdt =
F (s) + s2
(∫ f (t )dt )
t =0
s
2
Wאא
(∫∫ f (t )dtdt )
+
t =0
s
Wאאא
E
Wsאא
E
L( f (t − T ) = e − sT F ( s )
L(e at f (t )) = F ( s − a ) L( f ( at )) =
E
Wא
1 s F( ) a a
E
Wאאא E
אא lim sF ( s ) א s →∞
Wאf(t)אא
f (0) = lim sF ( s ) s →∞
-٩-
אא
א٢٦٢
א
EF
Wאא E אא lim sF ( s ) א s→0
Wאf(t)א
lim f (t ) = lim sF ( s) t →∞ s →0
WE١ J٢F Wאאאא
A t≥0 ⎩0 t < 0
x(t ) = ⎧⎨
Wא W،א ∞
X ( s ) = ∫ x (t )e
− st
0
∞
dt = ∫ Ae − st dt 0
A X ( s ) = − e − st s
∞ = 0
A s
u(t)،A=1אא Wא،א U ( s) =
1 s
WE٢ J٢F Wאא ⎧
x(t ) = ⎨ Ae
at
⎩ 0
t≥0 t<0
Wא Wאא ∞
∞
∞
X ( s ) = ∫ x(t )e dt = ∫ Ae e dt = ∫ Ae − ( s − a )t dt − st
0
at − st
0
A − st X ( s) = − e s−a
- ١٠ -
0
∞ = 0
A s−a
אא
א٢٦٢
EF
א
Wאאא ٤ J٢ אאאE١ J٢Fא
אאא،אא Wאא
אא
אאא
1
r (t ) = δ (t )
אא
A s
⎧ A; t ≥ 0 u (t ) = ⎨ ⎩ 0; t < 0
אא
A s2
⎧ At ; t ≥ 0 r (t ) = ⎨ ⎩ 0; t < 0
אאא
A s3
r (t ) = At 2
אא
A s+a
Ae− at
אאא
Aω s + ω2
A sin ωt
אאא
As s + ω2
A cos ωt
אא
2
2
n! (s + a )n +1
ω
t n e − at
(n = 1,2,3,....)
(s + a )2 + ω 2
e − at sin ωt
s+a (s + a )2 + ω 2
e − at cos ωt
Kאא،E١ J٢F
- ١١ -
אא
א٢٦٢
א
EF
Wא ٥ J٢ ،אא אא، אאאאא ، אאאאאא
Kאאא
F ( s) = F1 ( s) + F2 ( s) + ......... + Fn ( s) Wאאא f (t ) = f1 (t ) + f 2 (t ) + ......... + f n (t ) Wא،F(s)אא F ( s) = B( s) A( s )
Ksאא A(s)אB(s)א Wאאאאא
F ( s) = B( s) = K ( s + Z1 )(s + Z 2 )...(s + Z m ) A( s)
( s + P1 )( s + P2 )...( s + Pn )
אא، א Pn, ..., P2, P1 Zm, …., Z2, Z1 n ،אא אא،א א
א א א א א אא א א Km Kאא
א Pn, ..., P2, P1 א א א א
א،، אא אא אא א א Kא א
Kאאא،
Wאאא א١ J٥ J٢ Wאאא F ( s) = B( s) = A( s )
a1 a a + 2 + ..... + n s + P1 s + P2 s + Pn
אאKאא an ,...., a2 , a1 אא
Wאא
- ١٢ -
אא
א٢٦٢
א
EF
a1 = (F ( s)(s + P1 ) ) s = − P
1
a2 = (F (s)(s + P2 )) s = − P
2
KKKKKKKKKKKKKK an = (F (s)(s + Pn ) ) s = − P
n
WF(s)אאא f (t ) = a1e− P t + a2e− P t + ..... + ane− P t 1
2
n
WE٣ J٢F Wאאא
F (s) =
s+2 s(s + 1)(s + 3)
Wא s=-3, s=-1, s=0א F ( s) = a1 + s
a1 = (sF ( s) ) s = 0 = 2 , 3
a2 a + 3 s +1 s + 3
1 a2 = ((s + 1)F ( s) ) s = −1 = − , 2
Wאא a3 = ((s + 3)F ( s) ) s = −3 = −
1 6
WאF(s)אא F (s) =
2
1 1 3− 2 − 6 s (s + 1) (s + 3)
WאאאE١ J٢Fאאא f (t ) = 2 u (t ) − 1 e−t − 1 e−3t 3
2
6
Wאאאא א٢ J٥ J٢ ،א P2, P1אאא F ( s) = B( s) =
α1 s + α 2
Wאאאא
+
a3 a + ..... + n s + P3 s + Pn
(s + P1 )(s + P2 ) W s = − P1 א (s + P1 )(s + P2 ) א α 2 , α1s + α 2 s = − P = F ( s)(s + P1 )(s + P2 ) s = − P A( s)
1
1
- ١٣ -
α1
אא
א٢٦٢
א
EF
אאאא،אP1א
، אאאאאאאאא K α 2 ,
α1
WE٤ J٢F Wאא (s + 1)
F (s) =
s (s 2 + s + 1)
Wא Wאא
F ( s) = α21s + α 2
s + s +1
+
a s
Wאא s = 0,
s = −0.5 ± j 0.866
W
P3 = 0, P1 , P2 = −0.5 ± j 0.866 W s = −0.5 ± j 0.866 א (s 2 + s + 1) א s +1 s
s = −0.5 − j 0.866
= α1s + α 2 s = −0.5 − j 0.866
0.5 − j 0.866 = α1 (− 0.5 − j 0.866) + α 2 − 0.5 − j 0.866
0.5 − j 0.866 = α1 (0.25 + j 0.866 − 0.75) + α 2 (− 0.5 − j 0.866) Wאאאאאא
− 0.5α1 − 0.5α 2 = 0.5
0.866α1 − 0.866α 2 = −0.866 ∴ α1 + α 2 = −1
α1 − α 2 = −1 α 2 = 0 , α1 = −1 s(s + 1) =1 s (s 2 + s + 1) s = 0
a = sF ( s) s = 0 = F (s) =
−s 1 + s + s +1 s 2
- ١٤ -
Wאא
אא
א٢٦٢
א
EF
=
s + 0.5 1 0.5 − + 2 2 2 s (s + 0.5) + (0.866) (s + 0.5) + (0.866)2
אא، א אאאE١ J٢Fאאא Wאאא
f (t ) = 1 − e−0.5t cos(0.866t ) + 0.578e−0.5t sin (0.866t )
Wאאא א٣ J٥ J٢ אאא،אא rאאא Wאאאא،אא
F ( s) = B( s) = A( s)
br br −1 b1 + + ... + r r −1 (s + P1 ) (s + P1 ) (s + P2 )
Wאאא b1 ,...., br −1 , br אא
W، s = P1 (s + P1 )r א br = (s + P1 )r F ( s) s = − P
E
1
،sאא (s + P1 )r א E W، s = P1
br −1 =
d (s + P1 )r F ( s) ds s = − P1
W،sאEFאאא br − 2
E
1 d2 (s + P1 )r F (s) = 2 2 ds s=−P
1
אאאא
WאאKאא
br − k =
E
1 dk ( s + P1 ) r F ( s ) k! ds k s=−P
1
WE٥ J٢F Wאא
s 2 + 2s + 3 F (s) = (s + 1)3
Wא Wאאאאאא - ١٥ -
אא
א٢٦٢
א
EF
F (s) = K s = −1,
s = −1,
b3 b2 b + + 1 3 2 (s + 1) (s + 1) (s + 1)
s = −1 ،אאא
Wאא
b3 = F ( s)(s + 1)3 s = −1 = s 2 + 2s + 3 s = −1 = 2
(
b2 = 1
d 2 s + 2s + 3 1! ds
) s = −1
= 2 s + 2 s = −1 = 0
b1 = 1
d (2s + 2) = 2 = 1 2! ds 2 s = −1
WF(s)אאאא
F (s) =
2 1 + 3 (s + 1) (s + 1)
WאאאאE١ J٢Fאאאאא f (t ) = t 2e −t + e−t
for
t≥0
Wאאא ٦ J٢ א אא
אאאאאאאא،א א، אאאאאא،אא WאKאא an d
dy (t ) d n −1 y (t ) y (t ) + + ...... + a1 + a0 y (t ) = x(t ) a n −1 n −1 n dt dt dt
n
Fאא،אKKKKKאא،y(t)אאא
אאKE אאאא
Wא،אאאא Y ( s) = B( s) A( s )
אאאאאאא Wאאא y (s) = L−1⎛⎜⎜ B( s) ⎞⎟⎟ ⎝ A( s ) ⎠
- ١٦ -
אא
א٢٦٢
א
EF
WE٦ J٢F Wאאא
d 2 y (t ) dy (t ) +4 + 5 y (t ) = 5 x(t ) 2 dt dt dy (0) =2 y (0) = 1, dt
Wא א אאאאאאא W
⎛⎜ s 2Y ( s) − sy (0) − dy(0) ⎞⎟ + 4(sY ( s) − y (0) ) + 5Y ( s) = 5 X ( s) ⎝
dt ⎠
(s 2Y ( s) − s − 2) + 4(sY ( s) − 1) + 5Y ( s) = 5 X (s) (s 2 + 4s + 5)Y ( s) = 5 X ( s) + s + 6
X ( s) = 1 Wאאאאx(t)א s
Wא Y (s) =
s 2 + 6s + 5 s ( s 2 + 4 s + 5)
s = −2 ± Y (s) = a1 = 0 ,
j, s = 0
Wאא
a1s + a2 a + 3 2 (s + 2 ) + 1 s
a2 = 2 , a3 = 1
W
Wאאא
Wאאאא Y (s) = 1 + s
2 (s + 2)2 + 1
WאאאאE١ J٢Fאאאאא y (t ) = 1 + 2e −2t sin t
- ١٧ -
t≥0
אא
א٢٦٢
א
EF
W Wאא K١ t≥0،x(t) = 10
E
t≥0،x(t) =-3t
E
t≥0،z(t)=e5t
E
t≥0،x(t) = -10 E t≥0،x(t) = 2t
E
WE١ J٢Fאא t≥0،z(t)=10e-7t
E
t≥0،y(t)=10Cos5t
E
t≥0،w(t)=eatsinωt
E
K٢
t≥0،x(t)=2sin3t E
t≥0،v(t)=eatCosωt
E
Wאאא 1 s (s + 1)(s + 2) s+3 F (s) = (s + 1)(s + 2)2 13 F ( s) = 2 s (s + 4s + 13) F (s) =
2 y '' (t ) + 7 y ' (t ) + 3 y (t ) = 0
y ''' (t ) + 4 y '' (t ) + y ' (t ) + 3 y (t ) = 0
- ١٨ -
, y (t ) = 0 ,
K٣
E E E
Wאאא , y (t ) = 0 , y ' (0) = 0 E y ' (0) = 0 , y '' (0) = 0 E
K٤
EF
א
א
٣
אא
א٢٦٢
א
א
EF
Wאא Wא
Kא
K١
Kאא
K٣
Kאא
Kאא
- ١٩ -
K٢
K٤
אא
א٢٦٢
א
א
EF
W ١ J٣ א، אאאאא
אאאאאאאאא ، אאK אא
אKאאאאאא
אאאאאא א
אאK א א א K אאאא
אאאאאא Kאאאאא
אאא אא
K אא א ، אאאא،א
אאאאאאאאא
Kאאאאאאאא
، אאאא
Kאאאאאאאא
Wאא ٢ J٣ אאאאאא
אאאאאK W
an d
dy (t ) d n −1 y (t ) y (t ) + + ...... + a1 + a0 y (t ) = Kx(t ) a n −1 n −1 n dt dt dt
n
WאאKאy(t)אx(t) an s nY ( s) + an −1s n −1Y ( s) + ..... + a1sY ( s) + a0Y ( s) = KX ( s) אא אא אא G ( s) = Y ( s)
X ( s)
Wאא
=
K an s + an −1s + ... + a1s + a0 n
- ٢٠ -
n −1
אא
א٢٦٢
א
א
EF
WE١ J٣F Wאאאאאאאאאא 5 y '' (t ) + 2 y ' (t ) + y(t ) = 3x(t )
Wא Wאאאא 5s 2Y (s) + 2sY ( s) + Y ( s) = 3 X (s) Wא
(5s 2 + 2s + 1)Y ( s ) = 3 X ( s )
Y (s) 3 = 2 X ( s ) 5s + 2 s + 1
G ( s) =
Wא א٣ J٣ K א،אא
א،אאאא Kא
Wאאא א١ J٣ J٣ KאאאE١ J٣Fא א
J
אא
אא H
אא
H
א
א
א
א
Kאאאא،E١ J٣F Wאאאאא KאWא - ٢١ -
E
אא
א٢٦٢
א
א
EF
KEאאFאאWא אE KאאאWא
אאWא Kאא
KW
KאאEFWא
E E E
E
Wאאא ٢ J٣ J٣ אאאאאאא
K אאאאאאא Wאא
Wאאא אא אG1(s)אא،
E
KE٢ J٣F،G2(s)א
X(s)
G1(s)
Y(s)
G2(s)
=
X(s)
G(s)
Y(s)
Kאא،E٢ J٣F
Wאאא G ( s) = G1 ( s) ⋅ G2 ( s)
X(s)
אאא، G2 ( s) =
1 s +1
10 2s + 1
WE٢ J٣F Wאאא Y(s)
10 G1 ( s) = 1 ،אאא 2s + 1 s +1
Wא
Wאא
- ٢٢ -
אא
א٢٦٢
א
א
EF
G ( s) =
Y (s) = G1 ( s ) ⋅ G2 ( s ) X ( s) 1 10 = ⋅ s + 1 2s + 1 10 = 2 2 s + 3s + 1
Wאאא
10 2s + 3s + 1
X(s)
Y(s)
2
Wאאא E אG1(s)אא، אא KE٣ J٣F،G2(s)א
G1(s)
Y(s)
+
X(s)
=
X(s)
G(s)
Y(s)
G2(s)
Kאא،E٣ J٣F
Wאאא G ( s) = G1 ( s) + G2 ( s)
20 s+5
+
X(s)
אאא، G2 (s) =
WE٣ J٣F Wאאא
Y(s)
10 s +1
10 G1 ( s) = 20 ،אאא s +1 s+5
Wא
Wאא - ٢٣ -
אא
א٢٦٢
א
א
EF
G ( s) =
Y (s) = G1 ( s ) + G2 ( s ) X ( s) 20 10 = + s + 5 s +1 20( s + 1) 10( s + 5) = + ( s + 5)( s + 1) ( s + 5)( s + 1) 30s + 70 = ( s + 5)( s + 1) 30s + 70 = 2 s + 6s + 5
Wאאא X(s)
30 s + 70 s 2 + 6s + 5
Y(s)
WEאאאFאאא E א،
،H(s) אאאאG(s) אאא
KE٤ J٣F
X(s)
+
Y(s)
G(s)
-
=
X(s)
G(s)
Y(s)
H(s)
K،E٤ J٣F
Wאאא T ( s) =
G (s) 1 + G ( s) H ( s)
WE٤ J٣F Wאאא X(s)
+
-
1 0.25s + 1
- ٢٤ -
Y(s)
אא
א٢٦٢
א
א
EF
Wא
א،H(s)=1 G ( s ) =
1 ، א 0.25s + 1
Wאאאא
T ( s) =
Y ( s) G ( s) = X ( s) 1 + G ( s) H ( s) 1 = 0.25s + 1 1+ 1 0.25s + 1 1 = 0.25s + 2
Wאאא X(s)
1 0.25s + 2
- ٢٥ -
Y(s)
אא
א٢٦٢
א
א
EF
W Wאאאאאאאאא K١ y ' (t ) + 2 y (t ) = x(t ) E y '' (t ) + 20 y ' (t ) + 6 y (t ) = 9 x(t ) E 4 y ''' (t ) + 5 y '' (t ) + 2 y ' (t ) + y (t ) = 3x(t ) E y ''' (t ) + 3 y '' (t ) + 6 y ' (t ) + 2 y (t ) = x(t ) E Wאא K٢ X(s)
3 2s + 1
5 3s + 3
Y(s)
Wאא
K٣
Wאא
K٤
1 s+5
+
X(s)
Y(s)
2 s +1
X(s)
+
-
4 3s + 2
2 s
- ٢٦ -
Y(s)
EF
אאא
אאא
٤
אאא אאא
א٢٦٢
א
EF
Wאא Wא
Kאאאא
K١
Kאאאאא
K٢
Kאאאא
K٤
Kאאאאאא Kאא
- ٢٧ -
K٣
K٥
אאא אאא
א٢٦٢
א
EF
W ١ J٤ K אאאאאאא ،אאאאא
،אאאאKאא אFאא،א
KאאEא
אא،אאאאא
אאK אאאא
Kאא،א Wאא ٢ J٤
Wאאא א١ J٢ J٤ Wאאאאא τy ' (t ) + y (t ) = Kx (t )
W
KאאWτ •
KאאאWK • KאWx(t) •
KEאאFאWy(t) •
Wאאאא
אאK1y(t)א
Kאא
E
KKאx(t)א E
Kτאאy'(t)א
E
Wאאאאא א٢ J٢ J٤ אאאאאאא
،א،א، אא
W
- ٢٨ -
אאא אאא
א٢٦٢
א
EF
τsY ( s ) − y0 + Y ( s ) = KX ( s ) Wאא،אאאאא (τs + 1)Y ( s) = KX ( s) Wאאאא Y (s)
X ( s)
=
K τs + 1
WE١ J٤F Wאא 0.1y ' (t ) + y (t ) = 2 x(t ) y (0) = 0
Wא
Kאא
E
Kאא
E
Kא E
Wא Wy'א אE τ = 0 .1
Wx(t)א E K =2
Y (s)
X ( s)
=
Wאא
K 2 = τs + 1 0.1s + 1
E
WE٢ J٤F Wאאא R + vR(t) -
+
+
C
vin(t)
vout(t)
-
-
- ٢٩ -
אאא אאא
א٢٦٢
א
EF
Wא אאא،אRCאא
Wאאאאא،א Wאאאא vR (t ) + vout (t ) = vin (t )
Wאאאאאא vR (t ) = Ri(t ) אאאאאא i(t) ،Evc(t)=vout(t)Fאאאא
Wא
i(t ) = Cvout ' (t ) WאאvR(t) vR (t ) = RCvout ' (t ) WאאאvR(t) RCvout ' (t ) + vout (t ) = vin (t ) אRCאאאאא
אאK אא
אאאא אא
Wאא
Kvin(t)x(t)א •
Kvout(t)y(t)EאאFא • KRCא τ א• א
K1 Kאאא •
Wאאאאאאאאאא - ٣٠ -
אאא אאא
א٢٦٢
א
EF
Vout ( s) = Vin ( s )
1 RCs + 1
WE٣ J٤F ؟אאא qi(t)
R
h(t)
qo(t)
A
אAאא א K אאKRא
Wא א אאאאאא Wאאאאאאא qi ( t ) = q s ( t ) + qo ( t )
W
KאאאWqi(t) •
، אאא،אאאאWqs(t) • K q s (t ) = A dh(t ) Wאאא dt
qo ( t ) =
1 h(t ) ،אאאWqo(t) R
•
Wאא
A dh(t ) + 1 h(t ) = qi (t ) dt
R
WאאRא AR dh(t ) + h(t ) = Rqi (t ) dt
AR א א א
K R
- ٣١ -
אאא אאא
א٢٦٢
א
EF
Wאא ٣ J٤ Wאאא א١ J٣ J٤ Wאאאאא y" (t ) + 2αy' (t ) + ωo 2 y (t ) = Kωo 2 x(t )
W
K??אאWω0 •
K??אWα •
KאאאWK • Wאאאאא א٢ J٣ J٤ ،א،אא W،אאאאאא،א
s 2Y (s) + 2αsY (s) + ω02Y (s) = Kωo 2 X (s) Wאא ( s 2 + 2αs + ωo 2 )Y ( s) = Kωo 2 X (s) Wאאאא G ( s) = Y ( s)
X ( s)
Kω o 2 2 s + 2αs + ωo 2
=
WE٤ J٤F Wאא
y '' (t ) + 2 y ' (t ) + 5 y(t ) = 4 x(t )
W
Kאאא
E
Kאא E
Wא א א אא E Wאאאאא ωo 2 = 5 ⇒ ωo =
5 ⇒ ωo = 2.236
- ٣٢ -
אאא אאא
א٢٦٢
א
EF
2α = 2 K ωo = 4 2
⇒K=
2 ⇒α =1 2 4 ⇒K= ⇒ K = 0.8 5
⇒α = 4
ωo 2
אא אא E G ( s) = Y ( s)
=
X ( s)
Wא
4 s + 2s + 5 2
WE٥ J٤F Wאאא L
R
+ vR(t) - + vL(t) Vin(t)
+
+
C
-
Vout(t)
-
Wא אאא،אRLCאא K אאאאא ،א
Wאאאא vR (t ) + vL (t ) + vout (t ) = vin (t )
Wאאאאאא vR (t ) = Ri(t ) Wאאא vL (t ) = L di(t ) = Li ' (t ) dt
אאאאאאi(t)
،Evc(t)=vout(t)Fאאאא
Wא
i(t ) = Cvout ' (t ) Wאאאא - ٣٣ -
אאא אאא
א٢٦٢
א
EF
vR (t ) = RCvout ' (t ) Wאאאאא vL (t ) = Li ' (t ) = LCvout '' WאאאvL(t)vR(t) LCvout (t ) + RCvout ' (t ) + vout (t ) = vin (t ) W LCא
vout (t ) + R vout ' (t ) + L
1 1 vout (t ) = vin (t ) LC LC
אRLCאאאאא
אאKאא
אאאאאא
Wאא
Kvin(t)x(t)א •
Kvout(t)y(t)EאאFא • ωo =
α=
R 2L
1 LC
Wא •
Wא •
K1 Kאאא •
אאאאאאאאאאא
Wאא
⎛ 1 ⎞ ⎜ ⎟ Vout ( s ) LC ⎠ ⎝ = 1 Vin ( s ) ⎛R⎞ s 2 + ⎜ ⎟s + LC ⎝L⎠
WE٦ J٤F אאאאא
אKBא،KSpringM KF(t)
- ٣٤ -
אאא אאא
א٢٦٢
א
EF
KSpring
F(t)
M
B
x(t)
0
Wא א،אאא W،אאאאא ∑ F =F (t ) − FK
− FB = M
d 2 x (t ) dt 2
FK=KSpring x(t)אאאאWFK • FB = B
dx (t ) אאWFB • dt
W M d
2
x(t ) dx(t ) +B + K Spring x(t ) = F (t ) 2 dt dt
Wא x '' (t ) +
K 1 B ' F (t ) x (t ) + Spring x(t ) = M M M
Wא ωo =
K Spring
Wא •
M 1 B α= Wא • 2M 1 K = W• א K Spring
- ٣٥ -
אאא אאא
א٢٦٢
א
EF
W Wאא K١ 0.1y ' (t ) + y (t ) = 2 x(t ) y (0) = 0
Wא
Kאא
E
Kאא
E
Kא E
10kΩאאאRCאא
K٢
10µFאRCאאא
K٣
Wאא
K٤
K20µFא
K0.083 אאאא
y" ( t ) + 2 y ' ( t ) + 2 y ( t ) = 2 x ( t )
W
Kאאא
E
Kאא E
אRLCאאאאאא
KC=40µF،L=4mH،R=25kΩא
- ٣٦ -
K٥
EF
א
א
٥
- ٥٧ -
אא
א٢٦٢
א
א
EF
Wאא Wא
Kאא
K١
Kאא
K٣
Kאא
- ٣٧ -
K٢
אא
א٢٦٢
א
א
EF
W ١ J٥ K אאאא אא
אאאא Kאאאא
Wא א٢ J٥ ، אא אא W،א
KאאאאאWאא
E
אאאאאW א
E
KאאWא
E
Kאאאאאאא W אE K
Wאא ٣ J٥ אאא W Kאא
Wאא ١ J٣ J٥ אKאאאאאE١ J٥Fא
אאאאאאא
אאאאK
אא K אאא
Kא
א אא
א
אא
א
Kאא،E١ J٥Fא - ٣٨ -
א
אא
א٢٦٢
א
א
EF
Wאאאאאאא
Kאאאאאא
E
א، אאא E K אאאאKאאא
Kאאאאאאאא
،א،א
KEKKK ،Fא،
E
א، אאא
،Kא K،،א
K א،אא
K אאא
E
אא، אאא K
א، K
Kאאאאא
Kאאאאאא
Wאא ٢ J٣ J٥ אאE٢ J٥Fא، אא אאאאאאאKאאא
אe(t) אKאא א
אאאK p(t)א Kאאאאא
- ٣٩ -
אא
א٢٦٢
א
א
EF
e(t)
r(t) +
-
א
b(t)
p(t)
אא
אאא
c(t)
א
א
Kאא،E٢ J٥Fא
W،אא
אאאאאאWאא
Kא
E
KאWא אE
KאאאWc(t)א
،אאאאאWr(t)אא Kא
KאאאWb(t)אא
FאאאWe(t)א
אK אאEאאא
Kא
אאWp(t)א
Kא
E E
E
E
E
Wאאאאאאא
، אאאאאאא
،אא אאאא KTd אאאTaא
K،אא، Ta ≥ Td אאאאאא
،אאאאאא،א Kאא،אא - ٤٠ -
E
אא
א٢٦٢
א
א
EF
K אE
Kאא
Kאאאא
E
E
אאא אאא
Kאאא، Kאאאאאאאא
- ٤١ -
אא
א٢٦٢
א
א
EF
W ؟אא אK١
؟אאאאאאא
K٢
؟אאאאא
K٣
؟אאאא
K٥
؟אאא
Wאא Kאא
E
Kr(t)אא
E
Ke(t)א
E
K٤ K٦
Kא אE
Kb(t)אא
Kp(t)א
E E
؟אאאאא
K٧
K אאאא
K٩
؟אאא
Kא؟E
K٨
K אאאא K١٠ Kא؟
- ٤٢ -
EF
אא
אא
٦
- ٥٧ -
אא
א٢٦٢
א
אא
EF
Wאא Wא
Kאאאא
K١
Kא
K٣
Kאאא
Kאאא
K٤
Kאאא
K٦
Kאאאאא
Kאאא
- ٤٣ -
K٢
K٥
K٧
אא
א٢٦٢
א
אא
EF
W ١ J٦ Kאאאא،אאאא אאאאאאאאאאא
אאאKאאאא Kאאא
אא
e(t)
r(t) +
-
א
b(t)
p(t)
ﻣﺤﻮل اﻹﺷﺎرة
اﻟﻤﺸﻐﻞ
ﻋﻨﺼﺮ اﻟﺘﺤﻜﻢ اﻟﻨﻬﺎﺋﻲ
אאא
c(t)
א
א
Kאאא،E١ J٦Fא
אאאאאE١ J٦Fא
א אאKא Kאא
Wא ٢ J٦ אא אאא אKTTLא
Kא א אאא
W א٣ J٦ אאאאאאא אאאאKאאא
Wא
- ٤٤ -
אא
א٢٦٢
א
אא
EF
Wא א١ J٣ J٦ אאאאא K אא،א،א Wאאא
Wאאא א א
E
א، אאא KE٢ J٦FאאKאא
K،E٢ J٦Fא
KאאאאאאE٣ J٦Fא
Kאאאא،E٣ J٦F
Wאאא אE אא אא،אLא،
אKאאא
J٦Fא، אאE J٤ J٦FאKא
אאאאאE J٤
אאאאאאא K א - ٤٥ -
אא
א٢٦٢
א
אא
EF
אא
אא
EF
EF
Kאאאא،E٤ J٦F
KאאאאאE٥ J٦Fא
Kאאאאא،E٥ J٦F
Wאאא אE Kאאאאאאא ،EFא א Kא אאאKאאא ،א ١٠ אאאE٦ J٦F K، ١٠٠٠ Kאא
Kאאאאא،E٦ J٦F - ٤٦ -
אא
א٢٦٢
א
אא
EF
Wא א٢ J٣ J٦ אא אאא אK אאאא J٦FאK אאאאEאאFאא ، אאאE٧ אא،אאא Kאא א
א
אא
אא
אא
K،E٧ J٦F
Wאא F = PA
W
KNאWF
E
KPaאWP E rKm2א، A = πr 2 WאאWA
E
Kא א Kאאאאאאאא KאאאאאאאאE٨ J٦F - ٤٧ -
אא
א٢٦٢
א
אא
EF
A
B
KאאKEF
K אKEF
KאאKEF
Kאאאא،E٨ J٦F
Wאא א٣ J٣ J٦ א אאאא K אאאאא אא
KאאאאאE٩ J٦Fא
א
אאא
אא
אא אא
Kאאאאא،E٩ J٦F
אא אאאאאאא
K אאאאאאא Wאא
KאאWאא
E
KאאאWאא אE - ٤٨ -
אא
א٢٦٢
א
אא
EF
Wאא ٤ J٦ K אאאאאאא
א،אאאאא
Kאא
E٧ J٦FאאK אאא
אאאאאאא אא Kאאא،
Kאא
Kאאא،E٧ J٦Fא
Wאאא
Kאאאא
E
Kאאא E
KאאאאאE٨ J٦Fאא
Kאאאא،E٨ J٦Fא - ٤٩ -
אא
א٢٦٢
א
אא
EF
W ؟אאא אK١
؟אאאאאא
K٢
؟אאא
K٤
؟אאאא
K٦
؟אא؟אאאא
K٨
؟א
؟אאא
؟אאאא
؟אאא
- ٥٠ -
K٣ K٥ K٧
K٩
א
א٢٦٢
א
EF
AC motor actuator analog armature automation block diagram bode diagram cascade characteristic Equation characteristics chart recorder closed loop compensator control system control valve controlled variable controller critical damping cutoff frequency damping DC motor delay Time derivative derivative Controller design digital
א
א
א
אא
אאאא
א
א
א
א٢٦٢
א
EF
disturbance dynamic error feedback feedback path final control element flow meter flow rate forward path frequency response gain gain crossover frequency gain margin hydraulic input integral integral Controller lag compensator laplace transform lead compensator level magnitude manual control matrix motor open loop oscilloscope
אא
،
אא
א
א
א
א
א
א٢٦٢
א
EF
output over damping overshoot parallel peak time performance permanent response phase crossover frequency phase margin phase shift pneumatic polynomial potentiometer process programmablelogiccontrol proportional proportional controller reference input resonance frequency response response curve rise time root sensor series set point settling time
א
א
א
א
אא
א
א אא
،א
א
א
אא
א
א
א
א
א
א،א
אא
א
א٢٦٢
א
EF
signal conversion signal processing simulation specification stability stability criteria step input stepper motor summing junction system tachometer take off point time constant time domain response transducer transfer function transient response two position control underdamping unit step unity feedback
א
א
א
אא
אא
א
א
،א
אא
א
א
א
א
א
אא
א٢٦٢
א
EF
אא 1.
Johnson, C. D. Process Control Instrumentation Technology, Prentice Hall, 2002
2.
Bateson, R. N. Introduction to Control Systems Technology, Prentice Hall, 2002
3.
Ogata, K. Modern control Engineering, Prentice Hall, 1997
4.
Dorf, R. C. and Bishop, R. H. Modern Control Systems, Addisson Wesley, 1998
5.
אא، אא، א
١٩٩١،אאא
א
א٢٦٢
א
EF
א ١ ١
אא Kאא
٢
K
٢
Kאאא
٣
Kאאאאאא
٦
K
٣
٤
١ ١ J١
٢ J١
Kאא
٣ J١
Kאא
٥ J١
٤ J١
٧
٢
٨
Kאא
K
٨
٨
١١
١٢
١٢
١ J٢
K
٢ J٢
Kאאא
٤ J٢
K
Kא
٣ J٢ ٥ J٢
Kאאא א١ J٥ J٢
١٣
Kאאאא א٢ J٥ J٢
١٦
Kאאא
٦ J٢
١٩
א Kאא
٣
٢٠
Kאא
١٥
١٨
٢٠ ٢١
٢١
Kאאא א٣ J٥ J٢ K
K
Kאא
١ J٣
٢ J٣ ٣ J٣
Kאאא א١ J٣ J٣
א
א٢٦٢
א
EF
٢٢
٢٦
Kאאא ٢ J٣ J٣ K
٢٧
אאא Kאא
٤
٢٨
Kאא
٢٨
٢٨
K
١ J٤
٢ J٤
Kאאא א١ J٢ J٤
٢٨
Kאאאאא א٢ J٢ J٤
٣٢
Kאאא א١ J٣ J٤
٣٢
٣٢
אא
٣ J٤
Kאאאאא א٢ J٣ J٤ K
٣٧
א Kאא
٥
٣٨
Kאא
٣٦
٣٨
٣٨
K
١ J٥
Kאא
٣ J٥
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٣٨
Kאא ١ J٣ J٥
٤٢
٤٣
אא Kאא
٦
٤٤
Kא
٣٩
٤٤
٤٤
٤٥
٤٧
Kאא ٢ J٣ J٥
K
١ J٦
Kא
٣ J٦
٢ J٦
Kא א١ J٣ J٦
Kא א٢ J٣ J٦
א
א٢٦٢
א
EF
٤٨
Kאא א٣ J٣ J٦
٤٩
Kאא
٤ J٦
٥١
א
٥٠
K
٥٥
אא