G (s)
=
where
L ( s )
E (s)
=
K G (s)
a
p
=
K K ( Bs
a
i
+K
)
o
(
o = 0.12 s ( s + 0.0325 ) s + 2.5675 s + 6667
(
2
= s 0.12 s + 0.312 s + 80.05 s + 26
G ( s) =
3
(
2
)
43.33( s + 500)
s s + 2.6 s +
1 0-37 Forward-path Transfer Function:
G ( s) =
K s K K Ki N
1
s J t L a s + ( Ra J t + L a Bt + K 1 K 2 J t ) s + R a Bt + K 1 K 2 Bt + K b K i + K K Ki Kt
1
2
1.5 10 K
7
G ( s) =
R amp Error Consta
S tate Diagram:
Transfer Functions:
m ( s)
=
Ea ( s )
L ( s)
=
E a (s )
m ( s)
=
Ea ( s )
L ( s)
E a (s )
=
(
)
Ki s + BL s + KL / Ra
2
J m J L s + ( K e J L + BL J L + BL J m ) s + ( J m K L + J L
0.0008
0.0009
0.0010
53.83
52.68
51.38
43.72
41.81
40.09
1.125
1.140
1.162
T he phase margin is at a maximum of 54.69 deg when T = 0.0006. The performance worsens
if the value of a is less than 1000.
1 0-9 (a) Forward-path Transfer Function:
G ( s) =
100 K P +
KI
s
s + 10 s + 100
2
For
K v = 10,
K v = lim sG ( s ) = lim s
s 0
s 0
100 ( K P s + K I )
(
s s + 10 s + 100
2
)
= K I = 10
Thus the forwa
Maximum overshoot = 20.8%
(b)
S elect a relatively large value for K
The closed-loop zero at s
D
= K P
and a small value for K
/K
D
P
dynamics are governed by the other closed-loop poles. Let K
The fo
6.5500E+00
(b) T ime Responses: x ( 0 ) =
0 .1
0
0
0
x ( 0 ) =
0 .1
0
0
0 , the initial position of
With the initial states
'
'
x 1 or y1 is preturbed downward
x 3 = y 2 is
from its stable equilibrium
Comparing these results with the part 1, the final values are approximately the same but the shape of responses is
closed to the first order system behavior. Then the system time constant is obviously
7
8
9
10
I-6 (a)
x( k
20.781250
32.171875
49.257813
74.886719
17
18
19
20
+ 2 ) x ( k + 1) + 0 .1 x ( k ) = u s ( k )
x (0)
=
1311.681641
1968.522461
2953.783691
4431.675781
=0.
x (1 )
Taking the z-tr
(c) L ( s ) =
(
K ( s + 1)
s s + 3s + 1
3
P
)
=1
P
=
2
For stability, Z = 0. 11 = ( 0.5 P + P 1 ) 180 = 450
o
For K > 0, 11 = 90
o
For K < 0, 11 = +90
o
o
The system is unstable.
.
when K > 1.
11 = 27
0.5 Hz
1 Hz
327
2Hz
5Hz
328
10Hz
50Hz
329
As frequency increases, the phase shift of the input and output also increase. Also, the amplitude of the
output starts to decrease when the frequency increas
c. -10 V:
13. Study of the effect of viscous friction:
303
As seen in above figure, two different values for B are selected, zero and 0.0075. We could change the
final speed by 50% in open loop system
11-3
a)
b)
c)
d)
50rad/sec
0.0795 seconds
2.5A. The current
When Jm is increased by a factor of 2, it takes 0.159 seconds to reach 63% of its steady state
speed, which is exactly twice the original ti