Lecture 7
February 4, 2014
Spring 2014
Woo Ho Lee whlee@uta.edu

• Lab#2: Identification of DC motor transfer function
– Location: NH250
– Feb. 4, Tuesday
• 101A (3:30-5:20PM)
• 102A (5:30-7:20PM)
– Feb. 5, Wednesday
• 103A (2:00-3:50PM)
• 104 (4:00-5:50PM)
• Class website: www.uta.edu/ee/ngs/ee4314_control
– Homework #1: Due by Feb. 6.
– Lab #1 report is due by Feb. 13.
– Lab #2 handout is posted.
Woo Ho Lee Control Systems EE 4314, Spring 2014
2

• TAs:
– Sajeeb Rayhan: Home work grading and office hours
• mdsajeeb.rayhan@mavs.uta.edu
• Office hours: Tue/Thu 10AM-12PM, Mon 4PM-6PM at NH250
– Corina Bogdan: Lab preparation & homework and report grading
• Email: ioanacorina.bogdan@mavs.uta.edu
• Office: NH250
– Joe Sanford: Lab lecture
• Email: joe.sanford@MAVS.UTA.EDU
• Office: NH250
Woo Ho Lee Control Systems EE 4314, Spring 2014

• Four Sessions (Total: 42 students)
 Session 101: Tue: 3:30PM-5:20PM (12 students)
 101A (6)
 101B (6)
 Session 102: Tue: 5:30PM-7:20PM (11 students)
 102A (6)
 102B (5)
 Session 103: Wed: 2:00PM-3:50PM (12 students)
 103A (6)
 103B (6)
 Session 104: Wed: 4:00PM-5:50PM (7 students)
Woo Ho Lee Control Systems EE 4314, Spring 2014
4

 Lab #2: NH250
– 101A and 102A: Feb. 4 (Tue)
– 103A and 104: Feb. 5 (Wed)
– 101B and 102B: Feb. 11 (Tue)
– 103B: Feb. 12 (Wed)
Tuesday
101 (3:30-5:20)
102 (5:30-7:20)
Wednesday
103 (2-3:50)
104 (4-5:50)
Woo Ho Lee Control Systems EE 4314, Spring 2014
5

101B 101A
X
X
Saad Akhtar
Sanjeeb Banjara
Asrat Beshah
Blake Farmer
Hawariya Gebremedhien
Nadim Giotis
Daniel Goodman
Leighlan Jensen
Kevin Oseguera
Prabesh Poudel
Eric Reiser
Caroline Storm
Woo Ho Lee Control Systems EE 4314, Spring 2014
X
X
X
X
X
X x
6

102B
Laury Arcos
Matthew Barboza
Monica Beltran
Victoria Brandenburg
Israel Fierro
John Fierro
Haile Fintie
Samuel Luce
Blen Mamo
Nisha Shrestha
Christopher Williams
Woo Ho Lee Control Systems EE 4314, Spring 2014
102A
X
X
X
X
X
X x
7

103A 103B
X Joshua Berry
Pasquier Biyo
Aaron Dyreson
Pursottam Giri
Prem Kattel
Gregory Martin
Bardia Mojra
Vihang Parmar
Abison Ranjit
Thyag Ravi
Sharad Timilsina
Hannah Vuppula
Woo Ho Lee Control Systems EE 4314, Spring 2014
X
X x
X
X
X
X
X
X
X
X
8

• Physics
– Law of motors: 𝐹 = 𝐵𝑙𝑖
• Convert electric energy ( i ) to mechanical work ( F )
– Law of generator: 𝑒 𝑡 = 𝐵𝑙𝑣
• Mechanical motion
 electric voltage
Where 𝐵 : strength of magnetic field 𝑙 : length of a coil 𝑣 : velocity of the conductor
𝐹 : Force acting on the conductor 𝑒(𝑡) : voltage across the conductor
Woo Ho Lee Control Systems EE 4314, Spring 2014

Magnetic Force on Current Carrying Wire
• Force F = 𝑙 × B I
I : current
B : strength of magnetic field 𝑙 : length of a wire that carries current I through a magnetic field 𝑙
Woo Ho Lee Control Systems EE 4314, Spring 2014

• Force F = B𝑙I
• Torque T = 2F𝑎 = 2B𝑙aI = KtI
• Torque constant
K t
= 2B𝑙a 𝑙
F
I
I
F
B a =radius of wire loop
Woo Ho Lee Control Systems EE 4314, Spring 2014

Torque in Permanent Magnet DC Motor
• Torque T = 2nB𝑙aI = KtI
• Torque constant
K t
= 2nB𝑙a n=5 n = number of loops
Woo Ho Lee Control Systems EE 4314, Spring 2014

•
• Find dynamic equations
Find transfer function
 𝑚 𝑣 𝑎

= 𝑑𝜃 𝑑𝑡
Woo Ho Lee Control Systems EE 4314, Spring 2014

Woo Ho Lee Control Systems EE 4314, Spring 2014

Woo Ho Lee Control Systems EE 4314, Spring 2014

• Force acting on moving mass
𝐹 = 𝐵𝑙𝑖 l=2a
 n n : number of turns a : radius of core
𝐹
Woo Ho Lee Control Systems EE 4314, Spring 2014

• Applying KVL
• Applying Newton’s law
Woo Ho Lee Control Systems EE 4314, Spring 2014

• Heat flow 𝑞 =
1
𝑅
(𝑇
1
− 𝑇
2
) q : heat energy flow (J/sec)
R : thermal resistance
T : temperature
• Relation between temperature of the substance and heat flow
𝑇 =
1
𝐶 𝑞
C : thermal capacity
Woo Ho Lee Control Systems EE 4314, Spring 2014

• Find the differential equations that determine the temperature in the room 𝑇
1
(four sides are thermally insulated)
𝑇
1
Woo Ho Lee Control Systems EE 4314, Spring 2014

• Find the differential equations that determine the temperature in the room 𝑇
1
(four sides are thermally insulated)
𝑇
1
=
1
𝐶
1
(−𝑞
1
− 𝑞
2
1
)=
𝐶
1
1
𝑅
1
𝑇
0
− 𝑇
1
1 1
+
𝐶
1
𝑅
2
𝑇
0
− 𝑇
1
1
=
𝐶
1
1
(
𝑅
1
+
1
𝑅
2
) 𝑇
0
− 𝑇
1
𝑇
1
Woo Ho Lee Control Systems EE 4314, Spring 2014

• Physics governing fluid flow
Continuity equation: 𝑚 = 𝑤 𝑖𝑛 where
− 𝑤 𝑜𝑢𝑡 m : fluid mass within the system ( 𝑚 = 𝜌𝑉 = 𝜌𝐴ℎ) w in
: mass flow rate into the system w out
: mass flow rate out of the system
Differential equation that governs the height of water
ℎ = 1 𝜌𝐴
(𝑤 𝑖𝑛
−𝑤 𝑜𝑢𝑡
) (1)
A: area of the tank 𝜌 : density of water h : height of water 𝑚 = 𝜌𝐴 ℎ
Woo Ho Lee Control Systems EE 4314, Spring 2014

• Fluid flow through an orifice
1 𝑤 𝑜𝑢𝑡
= 𝑝
1
− 𝑝𝑎 (2)
𝑅 where 𝑝
1
=  𝑔ℎ + 𝑝 𝑎
: hydrostatic pressure 𝑝 𝑎
: ambient pressure
• Substituting (2) into (1) gives
ℎ = 1
(𝑤 𝑖𝑛
−
1
𝑅 𝑝
1
− 𝑝𝑎 ) (3) 𝜌𝐴
• Linearization involves selecting the operating point 𝑝
1
= 𝑝 𝑜
+  𝑝 (4)
Woo Ho Lee Control Systems EE 4314, Spring 2014

• Substituting (2) into (1) gives 𝑤 𝑜𝑢𝑡
=
1
𝑅 𝑝 𝑜
1
− 𝑝𝑎 =
𝑅
 𝑝 + (𝑝 𝑜
=
= 𝑝 𝑜
−𝑝𝑎
𝑅 𝑝 𝑜
−𝑝𝑎
𝑅
 𝑝+ (𝑝 𝑜
−𝑝𝑎)
[1 +
2 (𝑝 𝑜
= 𝑝 𝑜
1
−𝑝𝑎
 𝑝
−𝑝𝑎)
] p o
−pa
R
−𝑝𝑎)
1 +
 𝑝
(𝑝 𝑜
−𝑝𝑎)
(5)
• Substituting (5) into (3) gives
 ℎ =
1 𝜌𝐴
(𝑤 𝑖𝑛
− 𝑝 𝑜
• Since
 𝑝 =
 𝑔
 ℎ
−𝑝𝑎
𝑅
[1 +
1
2 (𝑝 𝑜
 𝑝
−𝑝𝑎)
]) (6)
 ℎ =
−𝑔
2𝐴𝑅 𝑝 𝑜
−𝑝𝑎
 ℎ + 𝑤 𝑖𝑛
𝐴

− 𝑝 𝑜
−𝑝𝑎

𝐴𝑅
Woo Ho Lee Control Systems EE 4314, Spring 2014

• Find nonlinear differential equations relating the movement
 of the control surface to the input displacement x of the valve.
Fluid in
Fluid out
Input
Output
Woo Ho Lee Control Systems EE 4314, Spring 2014

• Flow goes inside of piston
𝑄
1
1
= 𝜌𝑅
1 𝑝 𝑠
− 𝑝
1 𝑥
• Flow come out of piston
𝑄
2
1
= 𝜌𝑅
2 𝑝
2
− 𝑝 𝑒 𝑥
• Continuity relation
𝐴 𝑦 = 𝑄
1
= 𝑄
2
A: piston area
• Force equation
𝐴 𝑝
1
− 𝑝
2
− 𝐹 = 𝑚 𝑦 m : mass of piston and attached rod
• Moment equation
𝐼 𝜃 = 𝐹𝑙 cos 𝜃 − 𝐹 𝑎 𝑑
I : moment of inertia of the control surface and attachment
• Kinematic relationship between
 and y
Woo Ho Lee Control Systems EE 4314, Spring 2014 𝑦 = 𝑙 sin 𝜃

• Mechanical system
– Newton’s 2 nd law (translation):

F = ma
– Newton’s 2 nd law (rotation):

M = I

– Hook’s law: F=kx
• Electrical system
– KCL (Kirchhoff’s current law):
 𝐼 in
=
 𝐼
– KVL (Kirchhoff’s voltage law ):

V out closed loop
=0
– Ohm’s law
• Electromechanical system
– Law of motors: 𝐹 = 𝐵𝑙𝑖
Convert electric energy ( i ) to mechanical work ( F )
– Law of generator: 𝑒 𝑡 = 𝐵𝑙𝑣
Mechanical motion
 electric voltage
– Torque developed in a rotor: T = 𝐾 𝑡 𝑖
– Back emf: 𝑒 = 𝐾 𝑒
𝜃 𝑚
Woo Ho Lee Control Systems EE 4314, Spring 2014

• Block Diagram Model:
– Helps understand flow of information (signals) through a complex system
– Helps visualize I/O dependencies
– Elements of block diagram:
• Lines: Signals
• Blocks: Systems
• Summing junctions
• Pick-off points
U(s)
H(s)
Y(s)
U1
+
U2
U1+U2
Transfer Function Summer/Difference
Woo Ho Lee Control Systems EE 4314, Spring 2014
U
U
Pick-off point
U

Three Examples of Elementary Block Diagrams
(a) Cascaded system G
1
(s)G
2
(s) (b) Parallel system G
1
(s)+ G
2
(s)
(b) Negative feedback system
Woo Ho Lee Control Systems EE 4314, Spring 2014
𝐺
1
(𝑠)
1 + 𝐺
1 𝑠 𝐺
2
(𝑠)

=
=
Woo Ho Lee Control Systems EE 4314, Spring 2014

=
=
Woo Ho Lee Control Systems EE 4314, Spring 2014

• Example: Simplify the block diagram
Woo Ho Lee Control Systems EE 4314, Spring 2014

Woo Ho Lee Control Systems EE 4314, Spring 2014