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ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise
Prove that 푧 = 3푥 + 4푦 + 5 is a linear equation.
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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 )
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ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise
Linearize 푦 = log 푥 at x = 1 and at x = 10.
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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
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ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise
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ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise
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ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise
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ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise
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ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
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ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise
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ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise
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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.
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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
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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.
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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
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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
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ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise
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ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise
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ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise Plot the root locus of
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ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise
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ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise Plot the Nyquist Diagram of the system
)1()3()()(
sssKsHsG
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ASIAN INSTITUTE OF TECHNOLOGY MECHATRONICS
Manukid Parnichkun
Class Exercise
Exercise
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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
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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)