exercises 1

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)