Download Control Systems EE 4314

Control Systems
EE 4314
Lecture 5
January 28, 2014
Spring 2014
Woo Ho Lee
[email protected]
Announcement on Jan. 28
• Textbook: Available yet at UTA bookstore
• Office hour: Tue/Thu 3:30 pm – 5:00 pm NH250 Lab
• TAs:
– Sajeeb Rayhan: Home work grading and office hours
• [email protected]
• Office hours: Tue/Thu 10AM-12PM, Mon 4-6PM
• Location: NH250
• Lab#1: MATLAB & Simulink sessions
– Any questions?
• Class website update: www.uta.edu/ee/ngs/ee4314_control
– Homework #1: Due by Feb. 6.
– Lab #1 handout is posted. Lab #1 report is due by Feb. 13.
Woo Ho Lee Control Systems EE 4314, Spring 2014
2
Electrical Circuits
• Electric circuits: resistors, capacitors, inductors, and OPAmp
• Basic laws governing electric circuits
– Ohm’s Law
– Kirchhoff’s current law (KCL, node law): The sum of currents
entering a node (𝐼in) is equal to the sum of currents leaving the
same node (𝐼out) .
𝐼in= 𝐼out
– Kirchhoff’s voltage law (KVL, loop law): The sum of voltage
around any closed loop in an electrical circuit is zero.
V closed loop=0
Woo Ho Lee Control Systems EE 4314, Spring 2014
3
Electrical Circuits
• Find Voltage V3 and current I3 across R3
Woo Ho Lee Control Systems EE 4314, Spring 2014
4
Electric Circuit
Woo Ho Lee Control Systems EE 4314, Spring 2014
Electrical Circuits
• LRC circuit:
– Find mathematical equations
– Find transfer function:
• 𝑣 = 𝑅𝑖
𝑑𝑣
• 𝑖 = 𝐶 (𝑣 =
• 𝑣=
𝑑𝑡
𝑑𝑖
𝐿 𝑑𝑡
1
𝐶
𝑖dt)
LRC circuit
Woo Ho Lee Control Systems EE 4314, Spring 2014
6
Electrical Circuits
• Applying Kirchhoff’s voltage law
Loop 1 
• 𝑣 = 𝑅𝑖
𝑑𝑣
• 𝑖 = 𝐶 (𝑣 =
• 𝑣=
𝑑𝑡
𝑑𝑖
𝐿 𝑑𝑡
1
𝐶
𝑖dt)
Loop 2 
Taking the Laplace transform,
Loop 1
Loop 2
Transfer function
LRC circuit
Woo Ho Lee Control Systems EE 4314, Spring 2014
7
Electrical Circuits
• Write the dynamic equations and find the transfer
function
• 𝑣 = 𝑅𝑖
𝑑𝑣
1
• 𝑖 = 𝐶 𝑑𝑡 (𝑣 = 𝐶 𝑖dt)
𝑑𝑖
• 𝑣 = 𝐿 𝑑𝑡
Woo Ho Lee Control Systems EE 4314, Spring 2014
8
Electric Circuit
Woo Ho Lee Control Systems EE 4314, Spring 2014
Electric Circuits
• Find the transfer function
Woo Ho Lee Control Systems EE 4314, Spring 2014
Electric Circuits
• Applying KVL
𝑖1 = 𝑖2 +𝑖3
𝑖3 =𝑖1 -𝑖2
• Taking Laplace transform
Woo Ho Lee Control Systems EE 4314, Spring 2014
𝑖3
Complex Impedances
• Transfer function
• Complex impedances
– Resistance: R
– Capacitance: 1/Cs
– Inductance: Ls
Woo Ho Lee Control Systems EE 4314, Spring 2014
Operational Amplifiers
•
•
•
•
Op Amp is used to amplify signals.
Output
K: differential gain ()
No current flows into either input
of the op-amp (Input impedance is
infinite)
• Output impedance is zero
Woo Ho Lee Control Systems EE 4314, Spring 2014
Operational Amplifiers
eo
• Obtain the transfer function
ei
Woo Ho Lee Control Systems EE 4314, Spring 2014
Operational Amplifiers
eo
• Obtain the transfer function e
i
• Since 𝑒0 = 𝐾 0 − 𝑒 ′
eo eo
′
𝑒 = ≅ =0
𝐾

Hence
• Transfer function
eo
ei
=
Woo Ho Lee Control Systems EE 4314, Spring 2014
Inverting amplifier
Operational Amplifiers
eo
• Obtain the transfer function
ei
Woo Ho Lee Control Systems EE 4314, Spring 2014
Operational Amplifiers
eo
• Obtain the transfer function
ei
′
𝑒
𝑒′
Since 𝐾 = 
eo
=
ei
Woo Ho Lee Control Systems EE 4314, Spring 2014
Noninverting amplifier
Operational Amplifiers
eo
• Obtain the transfer function
ei
Woo Ho Lee Control Systems EE 4314, Spring 2014
Operational Amplifiers
eo
• Obtain the transfer function
ei
Woo Ho Lee Control Systems EE 4314, Spring 2014
First order lag circuit
Mechanical – Electrical Analogies
• Force-Voltage analogy
Woo Ho Lee Control Systems EE 4314, Spring 2014
Mechanical – Electrical Analogies
• Analogous quantities
– force f  voltage v
– velocity v  current i
– displacement x  charge q
– damper b  resistor R
– spring k  capacitor 1/C
– mass m  inductor L
Woo Ho Lee Control Systems EE 4314, Spring 2014
Mechanical-Electrical Analogy
Woo Ho Lee Control Systems EE 4314, Spring 2014
Woo Ho Lee Control Systems EE 4314, Spring 2014
Mechanical – Electrical Analogies
• Force-Current
Woo Ho Lee Control Systems EE 4314, Spring 2014