ELE 639 Lecture Notes - Lecture 6: Step Response, Damping Ratio, Domain Analysis

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ELE/BME639 Course Notes Winter 2011
Page 122
6 Time Domain Analysis: Dynamic Response Part 1
6.1 First Order Systems
A first order system is described by the transfer function in Equation 6-1:
1
)(
s
K
a
s
K
sG dc
Equation 6-1
G(s) has only one pole, and no zeros. Its unit step response can be derived as shown in Equation 6-2:
 
as
K
s
K
ass
K
sG
s
sY
21
)(
1
)(
a
K
s
K
K
a
K
as
K
K
as
s
2
0
1


)(11)(11)(
11
)(
teKte
a
K
ty
asa
K
sa
K
sY
t
dc
at
Equation 6-2
Note that:

00
)(1
dc
t
dc
K
dt
dy
te
K
dt
dy
Equation 6-3
If the unit step input is used, the process DC gain and time constant can be evaluated directly from the
graph, as shown in the following example.
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ELE/BME639 Course Notes Winter 2011
Page 123
6.1.1 Example
Consider a plot of the response of a certain unknown process, shown in Figure 6-1. We would like to
derive a model for this unknown system, i.e. a transfer function that would give a response closest to that
of our system, let's call it (s). The response looks like an exponential rise with a non-zero slope at t=0,
and is therefore identified as the response of a first order process (system). As such, the response can be
described by the following equation:
)(11)( teKty t
dc
Figure 6-1 First Order System response
where the DC gain and the exponential Time Constant can be read off directly from the plot. Since
 1
and  
 ∙
, then  
 1.2. To read off the Time Constant, take a tangent to the exponential
rise and identify the cross-over of it with the steady state value of the output (here equal to 1.2). As
indicated in Figure 6-1, = 0.2 sec. Therefore:

)(112.1)(
)(112.1)(
5
2.0
tety
tety
t
t
The process model transfer function will be then:

  1
1.2
0.2  1 6
5
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Document Summary

6 time domain analysis: dynamic response part 1. A first order system is described by the transfer function in equation 6-1: sg. G(s) has only one pole, and no zeros. Its unit step response can be derived as shown in equation 6-2: Note that: dy dt dy dt t e. If the unit step input is used, the process dc gain and time constant can be evaluated directly from the graph, as shown in the following example. Consider a plot of the response of a certain unknown process, shown in figure 6-1. We would like to derive a model for this unknown system, i. e. a transfer function that would give a response closest to that of our system, let"s call it (cid:1833)(cid:3040)(s). The response looks like an exponential rise with a non-zero slope at t=0, and is therefore identified as the response of a first order process (system). As such, the response can be described by the following equation: ty.

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