Cst3

ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

Control Systems

Tutorial 3

ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

Today’s class

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

ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

Problem 1 Gears

ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

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.

ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

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

ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

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

ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

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

ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

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:

ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

1

212

2

1

21222

)(N

NTKDsJs

N

NTKDJ

Step 3. Equations

ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

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

ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

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

ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

Exercise 1

Find Transfer Function? )(

)(

1

2

sT

s

ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

Problem 2 electromechanical system

ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

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

Problem 2

ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

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

ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

DC Motor – Electromechanical System

Torque-CurrentEquation

mmmmm TDJ

Mechanical Equation

ambaaaa eKLiRi

Electrical Equation

atm iKT

ElectromechanicalEquation

ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

DC motor example

)()()()( sEKssLssIRsI abmaaaa

)()()(2 sTssDsJs mmmmm

)()( sIKsT atm

Equations in Frequency domain