exercises 1
TRANSCRIPT
ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise
Prove that 푧 = 3푥 + 4푦 + 5 is a linear equation.
ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise
Prove that 푧 = 3푥 + 4푦 + 5 is a linear equation. Solution
푧 = 3푥 + 4푦 + 5
푧 − 5 = 3푥 + 4푦
푚 = 3푥 + 4푦
푎푚 + 푏푚 = 3(푎푥 + 푏푥 ) + 4(푎푦 + 푏푦 )
푎푚 + 푏푚 = 푎(3푥 + 4푦 ) + 푏(3푥 + 4푦 )
ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise
Linearize 푦 = log 푥 at x = 1 and at x = 10.
ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise
Linearize 푦 = log 푥 at x = 1 and at x = 10. Solution
At x = 1,
푦 = log 푥
푦 ≈ log푥| +1
푥|∆푥
푦 ≈ 0 + ∆푥
At x = 10,
푦 = log 푥
푦 ≈ log 푥| +1
푥|∆푥
푦 ≈ 1 + 0.1∆푥
ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise
ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise
ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise
ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise
ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise
ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise
ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise
Consider the system shown below, determine the response when the input is represented by (a)푅(푡) = 1, (b)푅(푡) = 푡.
ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise
Consider the system shown below, determine the response when the input is represented by (a)푅(푡) = 1, (b)푅(푡) = 푡.
Solution
Closed loop transfer function is determined. 퐶푅 =
1푠 + 2
(a) 푅(푡) = 1
퐶(푠) =1
푠 + 2∙
1푠
= −0.5푠 + 2
+0.5푠
퐶(푡) = −0.5푒 + 0.5
(a) 푅(푡) = 푡
퐶(푠) =1
푠 + 2 ∙1푠 =
0.25푠 + 2 −
0.25푠 +
0.5푠
퐶(푡) = 0.25푒 − 0.25 + 0.5푡
ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise Consider the RLC circuit again when R = 6 kΩ, L = 1 kH, and C = 200 µF, determine the
response when the input voltage is (a) a ramp input with unity slope and (b) a sinusoidal
input with amplitude of unity at frequency of 10 rad/sec.
ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise Consider the RLC circuit again when R = 6 kΩ, L = 1 kH, and C = 200 µF, determine the
response when the input voltage is (a) a ramp input with unity slope and (b) a sinusoidal
input with amplitude of unity at frequency of 10 rad/sec.
Transfer function from voltage input to current output is determined.
퐼푉 =
푠퐿푠 + 푅푠 + 1/퐶
퐼푉
=푠
1000푠 + 6000푠 + 5000= 0.001
푠푠 + 6푠 + 5
(a) When the voltage input is a ramp input with unity slope,
퐼 = 0.001푠
푠 + 6푠 + 5∙
1푠
= 0.0011
(푠 + 6푠 + 5)푠
퐼 = 0.0010.2푠−
0.25푠 + 1
+0.05푠 + 5
푖(푡) = 0.001(0.2 − 0.25푒 + 0.05푒 )
0 1 2 3 4 5 6 7 8 9 100
1
x 10-4
ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
(b) When the voltage input is a sinusoidal input with amplitude of unity at frequency of 10
rad/sec,
퐼 = 0.001푠
푠 + 6푠 + 5∙
10푠 + 100
퐼 = 0.001−0.0752푠 + 0.4752
푠 + 100 −0.1푠 + 1 +
0.0862푠 + 5
푖(푡) = 0.001(−0.0752 cos(10푡) + 0.04752sin (10푡) − 0.1푒 + 0.0862푒 )
푖(푡) = 0.001(0.089sin (10푡 − 1)− 0.1푒 + 0.0862푒 )
0 1 2 3 4 5 6 7 8 9 10-1.5
-1
-0.5
0
0.5
1
1.5x 10
-4
ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise
ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise
ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise Plot the root locus of
ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise
ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise Plot the Nyquist Diagram of the system
)1()3()()(
sssKsHsG
ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise
ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise Plot the Bode Diagram when the open loop system is represented by
)1()3()()(
sssKsHsG )1( K
ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise Plot the Bode Diagram when the open loop system is represented by
)1()3()()(
sssKsHsG )1( K
-40
-20
0
20
40
60
80
Mag
nitu
de (d
B)
10-2
10-1
100
101
102
-270
-225
-180
-135
-90
Phas
e (d
eg)
Bode Diagram
Frequency (rad/sec)