ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

Control Systems

Tutorial 3

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Today’s class

• Rotational System with gears• Electromechanical systems• Equation To SFG

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Problem 1 Gears

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Rotational System with gears

• A toothed wheel that works with others to alter the relation between the speed of a driving system.

• Used to attain mechanical matching of motor to load.

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Gearing System – Position relationship

1. Distance travel by Gear 1 must equal distance travel by Gear 2

1

2

2

1

2211

r

r

rr

2. Ratio of radius between Gear 1 and Gear 2 is equal to ratio of number of teeth between Gear 1 and Gear 2

1

2

2

1

2

1

r

r

N

N

Transfer Function

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Gearing System – Torque relationship

1. Assume work generated by Gear 1 is equal to work consumed by Gear 2

2211

21

TT

WW

2. From previous result

2

1

2

1

1

2

2

1

2211

N

N

T

T

T

T

TT

Transfer Function

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1. Decide the input and the output2. Draw free body diagram of the inertia3. Convert time function to frequency-domain4. Obtain the transfer function

Modeling Steps:

Example 1

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Step 2: Input and Output Variables

Angular position of input shaft θ1(t)

Output variable:

Input variable:

Angular position of input shaft θ2(t)

Output variable:

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1

212

2

1

21222

)(N

NTKDsJs

N

NTKDJ

Step 3. Equations

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Step 4: Transfer Function – θ2(t) as output

Inertial J:

By Laplace Transform

Inertial J:

)()(

)(

)()()(

21

2

1

2

1

21222

2

KDsJsN

N

sT

s

N

NTsKssDsJs

1

21222 N

NTKDJ

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Step 4: Transfer Function – θ1(t) as output

Inertial J:

Gear system relationship

)(

1

)(

)(2

2

12

2

122

2

11

1

KN

NDs

N

NJs

N

NsT

s

1

21222 N

NTKDJ

2

112 N

N

1

212

2 )()(N

NTsKsDJs

1

21

2

11

2 )()(N

NT

N

NsKsDJs

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Exercise 1

Find Transfer Function? )(

)(

1

2

sT

s

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Problem 2 electromechanical system

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Find the transfer function X(s)/Ei(s) for the system shown above. Hint: find the electromechanical equation

Problem 2

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𝐸𝑖 (𝑠 )=𝑅 𝐼 1 (𝑠)+ 1𝐶𝑠 [ 𝐼1 (𝑠 )− 𝐼2 (𝑠 ) ]

Writing equations in frequency domain

𝑠 𝐿𝐼2 (𝑠 )+ 1𝐶𝑠 [ 𝐼 2 (𝑠)− 𝐼 1 (𝑠) ]=−𝐸𝑏 (𝑠 )=−𝑠 𝐾1𝑋 (𝑠)

Problem 2

𝑃 (𝑠 )=𝐾 2 𝐼2 (𝑠 )= (𝑀𝑠2+𝐹𝑠+𝐾 ) 𝑋 (𝑠 )

𝐼 2 (𝑠)=(𝑀𝑠2+𝐹𝑠+𝐾 ) 𝑋 (𝑠 )  

𝐾 2

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DC Motor – Electromechanical System

Torque-CurrentEquation

mmmmm TDJ

Mechanical Equation

ambaaaa eKLiRi

Electrical Equation

atm iKT

ElectromechanicalEquation

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DC motor example

)()()()( sEKssLssIRsI abmaaaa

)()()(2 sTssDsJs mmmmm

)()( sIKsT atm

Equations in Frequency domain