Introductions to Systems Analysis

1
Introductions to Systems Analysis
2
First Order Systems
3
RC Circuit
R
vc -
C
-
vs
4
Motion and Friction
f
?v
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1st Order Systems
General first order systems math representation
x(t) input y(t) output a, b constants
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Second Order Systems
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RLC Circuit
R
L
vc(t) -
-
C
vs(t)
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Spring-Mass-Damper
Spring constant k
Mass m
Force F
Friction coefficient ?
y
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2nd Order Systems
General 2nd order system math representation
x(t) input y(t) output a, b, c constants
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Summary Systems Behavior
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Questions
How do you solve the 1st and 2nd order
differential equations?
What did you learn in MTH 245?
Remember any technical terms?
Heard about the 3-step approach?
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Equation solutions

• yn(t) homogeneous solution (natural response).
• the solution that describes systems natural
  behavior.
• it is there forever.
• solve for yn(t) by setting the driving force to 0
• yf(t) particular solution (forced response).
• the solution that describes systems response to
  external driving force.
• it will eventually be gone when external force is
  gone.
• solve for yf(t) by choosing a special function
  that satisfy the entire equation
• Initial conditions y(t) has to satisfy initial
  conditions.

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Forced Response(Particular solution)
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Force Response yf(t)
A forced response takes the form of the driving
force (For both 1st and 2nd order systems)
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Natural Response (Homogeneous Solution)
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Natural Response 1st Order System
Equation
Solution
A is a constant to be determined by initial
conditions
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Example
Find the capacitor voltage vc as a function of
time. The input voltage of the circuit is
vs(t)5e3t u(t). The initial capacitor voltage is
0.
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Natural Response of 2nd Order Systems An Example
What is the natural response of 2nd order RLC
circuit voltage vc?
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Solution to Example Problem
? Damping factor
Let
?0 Resonance frequency
Assume
Substitute into homogeneous equation
Characteristics equation
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Characteristic Equation
The characteristics equation has two roots
Natural response of the equation
c1 and c2 are determined by initial conditions
vc(0) amd vc(0)
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Overdamped Case
If ?gt?0,
Overdamped case
vc(t)? 0 as t? 8
vc(t)
c1c2
t
0
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Critically Damped Case
If ??0,
Critically damped
vc(t)? 0 as t? 8
vc(t)
c1
t
0
1/?
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Underdamped Case
If ?lt?0,
Underdamped
vc(t)
C
t
0
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Example
Solve Example 8.4 in the text (P320)
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Reading and Homework

• Todays coverage 310-322
• Next Lecture 646-649
• Homework
• HW 15.1, P303, 7.52. Due Monday, 4/11
• HW 15.2, P356, 8.7. Due Wednesday, 4/13.