control system

1. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Control Systems
Tutorial 3

2. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Today’s class
• Rotational System with gears
• Electromechanical systems
• Equation To SFG

3. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Problem 1 Gears

4. 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.

5. 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

6. 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

7. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
1. Decide the input and the output
2. Draw free body diagram of the inertia
3. Convert time function to frequency-domain
4. Obtain the transfer function
Modeling Steps:
Example 1

8. 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:

9. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
1
2
12
2
1
2
1222
)(
N
N
TKDsJs
N
N
TKDJ



 
Step 3. Equations

10. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Step 4: Transfer Function – θ2(t) as output
Inertial J:
By Laplace Transform
Inertial J:
)()(
)(
)()()(
2
1
2
1
2
1
2
1222
2
KDsJsN
N
sT
s
N
N
TsKssDsJs





1
2
1222
N
N
TKDJ   

11. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Step 4: Transfer Function – θ1(t) as output
Inertial J:
Gear system
relationship
)(
1
)(
)(
2
2
1
2
2
12
2
2
1
1
1
K
N
N
Ds
N
N
Js
N
NsT
s




















1
2
1222
N
N
TKDJ   
2
1
12
N
N
 
1
2
12
2
)()(
N
N
TsKsDJs  
1
2
1
2
1
1
2
)()(
N
N
T
N
N
sKsDJs  

12. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Exercise 1
Find Transfer Function? )(
)(
1
2
sT
s

13. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Problem 2 electromechanical
system

14. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Find the transfer function X(s)/Ei(s) for the system shown above.
Hint: find the electromechanical equation
Problem 2

15. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
𝐸𝑖 𝑠 = 𝑅𝐼1 𝑠 +
1
𝐶𝑠
𝐼1 𝑠 − 𝐼2 𝑠
Writing equations in frequency domain
𝑠𝐿𝐼2 𝑠 +
1
𝐶𝑠
𝐼2 𝑠 − 𝐼1 𝑠 = −𝐸 𝑏 𝑠 = −𝑠𝐾1 𝑋(𝑠)
Problem 2
𝑃 𝑠 = 𝐾2 𝐼2 𝑠 = 𝑀𝑠2 + 𝐹𝑠 + 𝐾 𝑋(𝑠)
𝐼2 𝑠 =
𝑀𝑠2
+ 𝐹𝑠 + 𝐾 𝑋(𝑠)
𝐾2

16. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
DC Motor – Electromechanical System
Torque-Current
Equation
mmmmm TDJ   
Mechanical Equation
ambaaaa eKLiRi  
Electrical Equation
atm iKT 
Electromechanical
Equation

17. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
DC motor example
)()()()( sEKssLssIRsI abmaaaa  
)()()(2
sTssDsJs mmmmm  
)()( sIKsT atm 
Equations in Frequency domain